NDA 2024 Maths · Mathematics · Continuity and Differentiability · hard

Let f(x)=begincasesxsin 1/x,&x≠0\\[4pt] 0,&x=0endcases. 1. f is continuous at x=0. 2. f is differentiable at x=0.

Single correct option

Let

f(x)=\begin{cases}x\sin\dfrac1x,&x\ne0\\[4pt] 0,&x=0\end{cases}

. Consider the following statements:
1.

f

is continuous at

x=0

.
2.

f

is differentiable at

x=0

.
Which of the statements given above is/are correct?

Show answer and solution

Answer: (A) 1 only

Since

|x\sin\frac1x|\le|x|\to0

, we get

f(x)\to0=f(0)

, so

f

is continuous at 0 and statement 1 is true. The difference quotient is

\frac{f(h)-f(0)}{h}=\sin\frac1h

, which oscillates between

-1

and

1

and has no limit as

h\to0

, so

f'(0)

does not exist and statement 2 is false. Trap: continuity does not imply differentiability.

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