(B) To check the continuity of $f(x)$ at $x = 1$,we evaluate the left-hand limit $(LHL)$ and the right-hand limit $(RHL)$.$LHL$: $\lim_{x \to 1^-} f(x

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$f(x)$ is discontinuous at $x = 1$

(B) To check the continuity of $f(x)$ at $x = 1$,we evaluate the left-hand limit $(LHL)$ and the right-hand limit $(RHL)$.$LHL$: $\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (x^2) = (1)^2 = 1$.$RHL$: $\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (x + 5) = 1 + 5 = 6$.Since the left-hand limit $(1)$ is not equal to the right-hand limit $(6)$,the limit $\lim_{x \to 1} f(x)$ does not exist.Therefore,$f(x)$ is discontinuous at $x = 1$.

Class 12 Mathematics Continuity and Differentiability Continuity

Easy

If $f(x) = \begin{cases} x^2, & \text{when } x \le 1 \\ x + 5, & \text{when } x > 1 \end{cases}$,then

A
$f(x)$ is continuous at $x = 1$
B
$f(x)$ is discontinuous at $x = 1$
C
$\lim_{x \to 1} f(x) = 1$
D
None of these

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Continuity and Differentiability

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Continuity

Discuss the continuity of the function $f$ defined by $f(x) = \frac{1}{x}, x \neq 0$.

Easy

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Let $f(x) = \begin{cases} (x - 1)^{\frac{1}{2 - x}}, & x > 1, x \neq 2 \\ k, & x = 2 \end{cases}$. The value of $k$ for which $f$ is continuous at $x = 2$ is

DifficultJEE MAIN 2018

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If the derivative of the function $f(x) = \begin{cases} ax^2 + b & \text{if } x < -1 \\ bx^2 + ax + 4 & \text{if } x \geq -1 \end{cases}$ is continuous everywhere, then:

DifficultMHT CET 2026

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If the function $f(x) = \begin{cases} k_{1}(x-\pi)^{2}-1, & x \leq \pi \\ k_{2} \cos x, & x>\pi \end{cases}$ is twice differentiable,then the ordered pair $(k_{1}, k_{2})$ is equal to

DifficultJEE MAIN 2020

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If the function defined by $f(x) = \frac{\log (1+x)^{1+x}}{x^2} - \frac{1}{x}, x \neq 0$ is continuous at $x=0$,then $6 f(0)$ is equal to

DifficultAP EAMCET 2021

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If $f(x) = \begin{cases} x^2, & \text{when } x \le 1 \\ x + 5, & \text{when } x > 1 \end{cases}$,then

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Discuss the continuity of the function $f$ defined by $f(x) = \frac{1}{x}, x \neq 0$.

Let $f(x) = \begin{cases} (x - 1)^{\frac{1}{2 - x}}, & x > 1, x \neq 2 \\ k, & x = 2 \end{cases}$. The value of $k$ for which $f$ is continuous at $x = 2$ is

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