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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 $x$.
Solution :
Given Variables & Initial Setup
The function is defined as $g(x) = x - [x]$, where $[x]$ is the greatest integer function (floor function). We are required to investigate the continuity of $g(x)$ at any arbitrary integer $n$, where $n \in \mathbb{Z}$.
A function $g(x)$ is continuous at $x = n$ if and only if:
$\lim_{x \to n^-} g(x) = \lim_{x \to n^+} g(x) = g(n)$
Step 1: Evaluating the Function at $x = n$
By the definition of the greatest integer function, for any integer $n$, $[n] = n$ [By definition of the floor function for integers].
Substituting this into the function definition:
$g(n) = n - [n] = n - n = 0$
Step 2: Evaluating the Left-Hand Limit (LHL)
Let $x = n - h$, where $h > 0$ and $h \to 0$. As $x \to n^-$, $h \to 0^+$.
For $x$ in the interval $(n-1, n)$, the greatest integer $[x] = n - 1$.
$\lim_{x \to n^-} g(x) = \lim_{h \to 0} ((n - h) - [n - h])$
Since $n-h$ is slightly less than $n$, $[n - h] = n - 1$.
$\lim_{x \to n^-} g(x) = \lim_{h \to 0} (n - h - (n - 1)) = \lim_{h \to 0} (1 - h) = 1$
Step 3: Evaluating the Right-Hand Limit (RHL)
Let $x = n + h$, where $h > 0$ and $h \to 0$. As $x \to n^+$, $h \to 0^+$.
For $x$ in the interval $[n, n+1)$, the greatest integer $[x] = n$.
$\lim_{x \to n^+} g(x) = \lim_{h \to 0} ((n + h) - [n + h])$
Since $n+h$ is slightly greater than or equal to $n$, $[n + h] = n$.
$\lim_{x \to n^+} g(x) = \lim_{h \to 0} (n + h - n) = \lim_{h \to 0} (h) = 0$
Step 4: Comparison and Conclusion
| Condition | Value |
|---|---|
| LHL ($x \to n^-$) | $1$ |
| RHL ($x \to n^+$) | $0$ |
| Function Value $g(n)$ | $0$ |
Since $\lim_{x \to n^-} g(x) \neq \lim_{x \to n^+} g(x)$, the limit $\lim_{x \to n} g(x)$ does not exist [By the Limit Existence Theorem]. Therefore, the function $g(x) = x - [x]$ is discontinuous at all integral points $n \in \mathbb{Z}$.
Visual Representation of $g(x) = x - [x]$
Final Solution: The function $g(x) = x - [x]$ is discontinuous at every integer $n$ because the left-hand limit is $1$ and the right-hand limit is $0$, resulting in a jump discontinuity of magnitude $1$.
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 $f(x) = \begin{cases} x^3 - 3, & \text{if } x \le 2 \\ x^2 + 1, & \text{if } x > 2 \end{cases}$
- Q12: Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \begin{cases} x^{10} - 1, & \text{if } x \le 1 \\ x^2, & \text{if } x > 1 \end{cases}$
- 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 $x = 1$?
- 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
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