Question

Statement I Every differentiable function is continuous but converse is not true.
Statement II Function $f(x)=|x|$ is continuous.

• A. Statement I is true; statement II is true; statement II is a correct explanation for statement I
• B. Statement I is true; statement II is true; statement II is not a correct explanation for statement I
• C. Statement I is true; statement II is false
• D. Statement I is false; statement II is true

Answer 1

Answer: (B)

In above solution we have proved that, every differentiable function is continuous.
But the converse of the above statement is not true. We know that, $f(x)=|x|$ is a continuous function. Consider the left hand limit
$\displaystyle\lim _{h \rightarrow 0^{-}} \frac{f(0+h)-f(0)}{h}=\frac{-h}{h}=-1$
The right hand limit
$\displaystyle\lim _{h \rightarrow 0^{+}} \frac{f(0+h)-f(0)}{h}=\frac{h}{h}=1$

Since, the above left and right hand limits at 0 are not equal. $\displaystyle\lim _{h \rightarrow 0} \frac{f(0+h)-f(0)}{h}$ does not exist and hence $f$ is not differentiable at 0 . Thus, $f$ is not a differentiable function.