Find the best tutors and institutes for Class 12 Tuition
Q12:
Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \begin{cases} x^{10} - 1, & \text{if } x \le 1 \\ x^2, & \text{if } x > 1 \end{cases}$
Solution :
Step 1: Identifying the Domain and Potential Points of Discontinuity
The function $f(x)$ is defined in a piecewise manner over the set of all real numbers $\mathbb{R}$. The domain is partitioned at $x = 1$.
- For $x < 1$, $f(x) = x^{10} - 1$, which is a polynomial function. Polynomial functions are continuous everywhere in their domain [Per the continuity theorem for polynomials].
- For $x > 1$, $f(x) = x^2$, which is also a polynomial function, hence continuous for all $x > 1$.
- The only potential point of discontinuity is at the transition point, $x = 1$.
Step 2: Checking Continuity at $x = 1$
A function $f(x)$ is continuous at $x = c$ if and only if $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)$ [Definition of Continuity].
Evaluation of the Left-Hand Limit (LHL):
$\lim_{x \to 1^-} f(x) = \lim_{x \to 1} (x^{10} - 1)$
$= (1)^{10} - 1 = 1 - 1 = 0$
Evaluation of the Right-Hand Limit (RHL):
$\lim_{x \to 1^+} f(x) = \lim_{x \to 1} (x^2)$
$= (1)^2 = 1$
Evaluation of the Function Value at $x = 1$:
$f(1) = 1^{10} - 1 = 0$ [Using the condition $x \le 1$]
Step 3: Analytical Comparison
| Limit Type | Result |
|---|---|
| LHL ($\lim_{x \to 1^-} f(x)$) | $0$ |
| RHL ($\lim_{x \to 1^+} f(x)$) | $1$ |
| Function Value ($f(1)$) | $0$ |
Since $\lim_{x \to 1^-} f(x) \neq \lim_{x \to 1^+} f(x)$, the limit $\lim_{x \to 1} f(x)$ does not exist [By the Limit Existence Theorem]. Consequently, the function fails the criteria for continuity at $x = 1$.
Visual Representation of Discontinuity
Final Solution: The function $f$ is discontinuous at $x = 1$. There are no other points of discontinuity in the domain of $f$.
More Questions from Class 12 Mathematics Continuity and Differentiability EXERCISE 5.1
- Q1: Prove that the function $f(x) = 5x - 3$ is continuous at $x = 0$, at $x = -3$ and at $x = 5$.
- Q2: Examine the continuity of the function $f(x) = 2x^2 - 1$ at $x = 3$.
- Q3(a): Examine the following functions for continuity. (a) $f(x) = x - 5$
- Q3(b): Examine the following functions for continuity. (b) $f(x) = \frac{1}{x-5}$, $x \neq 5$
- Q3(c): Examine the following functions for continuity. (c) $f(x) = \frac{x^2 - 25}{x+5}$, $x \neq -5$
- Q3(d): Examine the following functions for continuity. (d) $f(x) = |x - 5|$
- Q4: Prove that the function $f(x) = x^n$ is continuous at $x = n$, where $n$ is a positive integer.
- Q5: Is the function $f$ defined by $f(x) = \begin{cases} x, & \text{if } x \le 1 \\ 5, & \text{if } x > 1 \end{cases}$ continuous at $x = 0$? At $x = 1$? At $x = 2$?
- Q6: Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \begin{cases} 2x + 3, & \text{if } x \le 2 \\ 2x - 3, & \text{if } x > 2 \end{cases}$
- Q7: Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \begin{cases} |x| + 3, & \text{if } x \le -3 \\ -2x, & \text{if } -3 < x < 3 \\ 6x + 2, & \text{if } x \ge 3 \end{cases}$
- Q8: Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \begin{cases} \frac{|x|}{x}, & \text{if } x \neq 0 \\ 0, & \text{if } x = 0 \end{cases}$
- Q9: Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \begin{cases} \frac{x}{|x|}, & \text{if } x < 0 \\ -1, & \text{if } x \ge 0 \end{cases}$
- Q10: Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \begin{cases} x + 1, & \text{if } x \ge 1 \\ x^2 + 1, & \text{if } x < 1 \end{cases}$
- Q11: Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \begin{cases} x^3 - 3, & \text{if } x \le 2 \\ x^2 + 1, & \text{if } x > 2 \end{cases}$
- Q13: Is the function defined by $f(x) = \begin{cases} x + 5, & \text{if } x \le 1 \\ x - 5, & \text{if } x > 1 \end{cases}$ a continuous function?
- Q14: Discuss the continuity of the function $f$, where $f$ is defined by $f(x) = \begin{cases} 3, & \text{if } 0 \le x \le 1 \\ 4, & \text{if } 1 < x < 3 \\ 5, & \text{if } 3 \le x \le 10 \end{cases}$
- Q15: Discuss the continuity of the function $f$, where $f$ is defined by $f(x) = \begin{cases} 2x, & \text{if } x < 0 \\ 0, & \text{if } 0 \le x \le 1 \\ 4x, & \text{if } x > 1 \end{cases}$
- Q16: Discuss the continuity of the function $f$, where $f$ is defined by $f(x) = \begin{cases} -2, & \text{if } x \le -1 \\ 2x, & \text{if } -1 < x \le 1 \\ 2, & \text{if } x > 1 \end{cases}$
- Q17: Find the relationship between $a$ and $b$ so that the function $f$ defined by $f(x) = \begin{cases} ax + 1, & \text{if } x \le 3 \\ bx + 3, & \text{if } x > 3 \end{cases}$ is continuous at $x = 3$.
- Q18: For what value of $\lambda$ is the function defined by $f(x) = \begin{cases} \lambda(x^2 - 2x), & \text{if } x \le 0 \\ 4x + 1, & \text{if } x > 0 \end{cases}$ continuous at $x = 0$? What about continuity at $x = 1$?
- Q19: Show that the function defined by $g(x) = x - [x]$ is discontinuous at all integral points. Here $[x]$ denotes the greatest integer less than or equal to $x$.
- Q20: Is the function defined by $f(x) = x^2 - \sin x + 5$ continuous at $x = \pi$?
- Q21(a): Discuss the continuity of the following functions: (a) $f(x) = \sin x + \cos x$
- Q21(b): Discuss the continuity of the following functions: (b) $f(x) = \sin x - \cos x$
- Q21(c): Discuss the continuity of the following functions: (c) $f(x) = \sin x \cdot \cos x$
- Q22: Discuss the continuity of the cosine, cosecant, secant and cotangent functions.
- Q23: Find all points of discontinuity of $f$, where $f(x) = \begin{cases} \frac{\sin x}{x}, & \text{if } x < 0 \\ x + 1, & \text{if } x \ge 0 \end{cases}$
- Q24: Determine if $f$ defined by $f(x) = \begin{cases} x^2 \sin \frac{1}{x}, & \text{if } x \neq 0 \\ 0, & \text{if } x = 0 \end{cases}$ is a continuous function?
- Q25: Examine the continuity of $f$, where $f$ is defined by $f(x) = \begin{cases} \sin x - \cos x, & \text{if } x \neq 0 \\ -1, & \text{if } x = 0 \end{cases}$
- Q26: Find the values of $k$ so that the function $f$ is continuous at the indicated point in Exercises 26 to 29. $f(x) = \begin{cases} \frac{k \cos x}{\pi - 2x}, & \text{if } x \neq \frac{\pi}{2} \\ 3, & \text{if } x = \frac{\pi}{2} \end{cases}$ at $x = \frac{\pi}{2}$
- Q27: Find the values of $k$ so that the function $f$ is continuous at the indicated point in Exercises 26 to 29. $f(x) = \begin{cases} kx^2, & \text{if } x \le 2 \\ 3, & \text{if } x > 2 \end{cases}$ at $x = 2$
- Q28: Find the values of $k$ so that the function $f$ is continuous at the indicated point in Exercises 26 to 29. $f(x) = \begin{cases} kx + 1, & \text{if } x \le \pi \\ \cos x, & \text{if } x > \pi \end{cases}$ at $x = \pi$
- Q29: Find the values of $k$ so that the function $f$ is continuous at the indicated point in Exercises 26 to 29. $f(x) = \begin{cases} kx + 1, & \text{if } x \le 5 \\ 3x - 5, & \text{if } x > 5 \end{cases}$ at $x = 5$
- Q30: Find the values of $a$ and $b$ such that the function defined by $f(x) = \begin{cases} 5, & \text{if } x \le 2 \\ ax + b, & \text{if } 2 < x < 10 \\ 21, & \text{if } x \ge 10 \end{cases}$ is a continuous function.
- Q31: Show that the function defined by $f(x) = \cos(x^2)$ is a continuous function.
- Q32: Show that the function defined by $f(x) = |\cos x|$ is a continuous function.
- Q33: Examine that $\sin|x|$ is a continuous function.
- Q34: Find all the points of discontinuity of $f$ defined by $f(x) = |x| - |x + 1|$.
CBSE Solutions for Class 12 Mathematics Continuity and Differentiability
- Continuity and Differentiability Class 12 EXERCISE 5.2
- Continuity and Differentiability Class 12 EXERCISE 5.3
- Continuity and Differentiability Class 12 EXERCISE 5.4
- Continuity and Differentiability Class 12 EXERCISE 5.5
- Continuity and Differentiability Class 12 EXERCISE 5.6
- Continuity and Differentiability Class 12 EXERCISE 5.7
Chapters in CBSE - Class 12 Mathematics
Top Tutors who teach Continuity and Differentiability
Top Tutor
Ram Nagar, Jalandhar
4 yrs of
Exp
7 students
218 hrs of classes completed
500
per hour
Online Classes
Tutor's home
Even if you have weak fundamental knowledge in chemistry at the beginning of class 12th, you don't have to be worried if you come under my tutelage. I have many students that feel scared from chemistry before they start taking my classes but I structure my course in such a way that they start enjoying Chemistry and their conceptual knowledge of class 11th also becomes good so that they can prepare for competitive exams easily.
A sincerity ,very professional and approachable attitude has helped my daughter to grasp the basics of class 12 chemistry and as my daughter was slightly behind the syllabus extra classes are being conducted to cover the syllabus. The cryptic and holistic notes also have helped a lot I am hopeful that my daughter will benefit immensely.
Top Tutor
Amrapali Dream Valley, Noida
10 yrs of
Exp
4 students
41 hrs of classes completed
500
per hour
Online Classes
Dear Parent/Student *Worried about 2022-23 Examinations*?? Mr. Ved Prakash is providing *Home tuitions*/*Online tuition of *Math & Applied Math* for class *XI and XII*. Have an edge over others by utilising your precious time. *Added Advantages:* 1. Notes and worksheet of every chapter will be provided. 2. Daily test and counselling. 3. Special work on problem solving skills. 4. 100 % result guaranteed. *Experience* - 10 years If interested contact Mr. Ved Prakash
It was an honour to have been Ved sir’s student. He is very thorough with his concepts, and does not miss out on any explanation when he is teaching something new, or even something familiar. He is patient, very knowledgeable in his field, and clear with what to teach and how to teach. Along with solving questions in class, he gives just enough homework, not too much or too less, so that the student has enough practice. I can certainly say that after just a few months of being taught by Ved sir, I started to love math again and show a substantial improvement. I would definitely recommend him to someone who wants to learn the basics, get more clear in a concept, or to even just have fun with maths.
Top Tutor
Ashok Nagar D' Souza Layout, Bangalore
5 yrs of
Exp
5 students
851 hrs of classes completed
500
per hour
Online Classes
Total Years of Experience: Over [X years] of experience in teaching across various academic levels, focusing on delivering engaging, student-centered education. 2. Educational Levels Taught: Primary, Secondary, and Senior Secondary levels Undergraduate-level classes (if applicable) Coaching for competitive exams (NEET, JEE, UPSC, etc.) 3. Key Responsibilities: Planning and delivering lessons based on prescribed curriculum. Assessing students’ understanding through regular tests, quizzes, and assignments. Using various teaching methodologies including lectures, group activities, demonstrations, and ICT tools. Providing remedial classes for weak students and enrichment activities for high performers. Maintaining discipline, fostering motivation, and ensuring an inclusive classroom environment. 4. Classroom Management: Successfully managed classrooms of 30–60 students. Promoted active participation and collaborative learning. Ensured timely syllabus completion while maintaining quality of instruction. 5. Communication and Interaction: Maintained regular communication with parents regarding student progress. Participated in parent-teacher meetings, academic planning sessions, and faculty training programs. 6. Technology Integration: Used smart boards, PowerPoint presentations, online quizzes, and digital simulations to enhance learning outcomes. Conducted online classes using platforms like Google Meet, Zoom, and Microsoft Teams during the COVID-19 pandemic. 7. Co-curricular and Administrative Roles: Organized and participated in educational tours, science exhibitions, and debate competitions. Assisted in exam invigilation, results compilation, and record keeping. 8. Achievements and Impact: Improved student performance and pass percentage year-on-year. Received positive feedback from students and parents for teaching effectiveness and approachability. Mentored students who went on to excel in competitive exams and higher education
had a great learning experience under the guidance of my History teacher. Their teaching style is clear, engaging, and well-structured. Complex topics like the Nationalism in India, Partition, and World War themes were explained with such depth and simplicity that it made understanding and remembering easy. They don’t just teach dates and events—they bring history to life with stories, critical thinking, and real-world connections. Regular assessments, personalized feedback, and doubt-clearing sessions really helped me improve my writing and analytical skills. Highly recommended for any Class 12th student aiming for high marks in board exams!
Top Tutor
Bhamian Khurd, Ludhiana
20 yrs of
Exp
2 students
12109 hrs of classes completed
300
per hour
Online Classes
Tutor's home
I am the author of seven books and a proud recipient of the Best Teacher Award from a reputed CBSE school. With over eight years of teaching experience at the school level and fifteen years of expertise in teaching Communication Skills and English Literature at a prestigious university, I bring a strong academic foundation and practical understanding of student development. My diverse teaching background has enabled me to mentor learners with varying abilities, helping them grow intellectually, linguistically, and personally. With this rich experience, I strive to ensure the all-around development of every student, fostering confidence, clarity of thought, and a lasting appreciation for the English language.
Super Tutor
Varthur, Bangalore
15 yrs of
Exp
11 students
497 hrs of classes completed
1000
per hour
Online Classes
I have over ten years of experience in the field of accounts. I am an IELTS-certified professional and have completed advanced Spoken and General English courses from the British Council. I hold a master's degree in Commerce and am interested in teaching. I have worked for Accenture, Deloitte, AIG, and a government-owned company, National Insurance. These experiences have given me real-time experience and expertise in the field of accounts, which I am eager to share with my students. As an IELTS 9 band scorer, I teach accounts and English to students in grades eleventh and twelfth. I offer Spoken English classes only online, one-on-one and in batches of up to tenth students. Each class lasts two hours a day, the introductory course lasts one and a half months, while the advanced course takes fifteen months to complete. For Spoken English, it takes around twenty to thirty hours to complete the basic level and forty-five to fifty hours for the advanced level. I accept students aged eleven years and above. I design a lesson plan that includes storytelling concepts, real-life examples, and practical exercises. I follow the Cambridge syllabus for accounts, and for English, I use practical-oriented techniques. I also frequently use a whiteboard, video, and audio links to help my students learn, and I provide soft copies of the materials, assignments, and question-and-answer sets for accounts.
The classes were extremely helpful, providing a strong foundation and boosting my confidence in the subject.
Other Subjects in CBSE - Class 12
Download free CBSE - Class 12 Mathematics Continuity and Differentiability EXERCISE 5.1 worksheets
Download Now
