Calculus

Continuity & Differentiability

Easy

core

Let $f(x) = |x-1| + |x-2|, x \in R$ . Then which of the following statements is/are TRUE?

(A) $f$ is continuous everywhere.

(B) $f$ is continuous everywhere except at $x = 1$ and $x = 2$

(C) $f$ is not differentiable at $x = 1$ and $x = 2$

(D) $f$ is differentiable at $x = 1$ and $x = 2$

Choose the correct answer from the options given below:

(A) and (D) only

(B) and (D) only

(A) and (C) only

(B) and (C) only

✅ Correct Option: 3

A sum of modulus functions is continuous on all of $\mathbb{R}$ , so (A) is true and (B) false. The graph has corners where each modulus changes sign, so the left and right derivatives differ at $x=1$ and $x=2$ ; hence (C) is true and (D) false.

2025: 22 May Shift 2

Match List-I with List-II

Where $\mathbb{R}$ is set of real numbers

List-I	List-II
(A) $\sin x$ is continuous on:	(I) $\mathbb{R} - \{0\}$
(B) $\tan x$ is continuous on:	(II) $\mathbb{R}$
(C) $\cot x$ is continuous on:	(III) $\mathbb{R} - \{n\pi: n \in \mathbb{Z}\}$
(D) $x^{-n}, n \in \mathbb{N}$ is continuous on:	(IV) $\mathbb{R} - \left\{(2n + 1)\frac{\pi}{2}: n \in \mathbb{Z}\right\}$

Choose the correct answer from the options given below:

2022: 4 Aug Shift 1

If $y = \frac{\sqrt{x+1} + \sqrt{x-1}}{\sqrt{x+1} - \sqrt{x-1}}$ , then $(x^2-1)^{3/2} \frac{d^2y}{dx^2} =$

2025: 26 May Shift 2

Derivative of $x^x$ with respect to $x\log x$ is

2025: 21 May Shift 1

If $x\sqrt{1 + y} + y\sqrt{1 + x} = 0$ , where $|x| < 1, |y| < 1$ and $x ≠ y$ , then

2026: 22 May Shift 1

If $y=\log_e\left(\frac{x^2}{e^2}\right)$ , then $\frac{d^2y}{dx^2}$ is equal to:

2026: 31 May Shift 2

If the function $f(x) = \dfrac{(4^x-1)^3}{\sin\dfrac{x}{4}\cdot\log\left(1+\dfrac{x^2}{3}\right)}$ is continuous at $x = 0$ , then the value of $f(0)$ is: (Take $\log_e x = \log x$ )

Solution

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