IGNOU MST 1 2026 - Discuss the continuity and differentiability of the following function

Question

Discuss the continuity and differentiability of the following function at $x = 5$ .

$$f(x) = \begin{cases} |x - 5|, & x \neq 5 \\ 0, & x = 5 \end{cases}$

19 Jan 2026

Answer :

Word Count : 304

We are asked to examine the continuity and differentiability of the function [ f(x) = \begin{cases} |x-5|, & x \neq 5 \ 0, & x = 5 \end{cases} ] at (x = 5). Let’s go step by step. --- Step 1: Check continuity at (x = 5) A function is continuous at (x = 5) if [ \lim_{x \to 5} f(x) = f(5). ] Here, (f(5) = 0). Now, calculate the limit from the left and right: * Left-hand limit ((x \to 5^-)): [ \lim_{x \to 5^-} f(x) = \lim_{x \to 5^-} |x-5| =

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