Question

Give an example of a function which is continuos but not differentiable at at a point.

Answer 1

Consider a function,  

`f(x) = {(x,, x,>,0),(-x,, x,le,0):}`

This mod function is continuous at x=0 but not differentiable at x=0.

Continuity at x=0, we have:

(LHL at x = 0)

\[\lim_{x \to 0^-} f(x) \]
\[ = \lim_{h \to 0} f(0 - h) \]
\[ = \lim_{h \to 0} - (0 - h) \]
\[ = 0\]

(RHL at x = 0)

\[\lim_{x \to 0^+} f(x) \]
\[ = \lim_{h \to 0} f(0 + h) \]
\[ = \lim_{h \to 0} (0 + h) \]
\[ = 0\]

and  

\[f(0) = 0\]

Thus, 

\[\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0) .\]

Hence, 

\[f(x)\] is continuous at 

\[x = 0 .\]

Now, we will check the differentiability at x=0, we have:

(LHD at x = 0)

\[\lim_{x \to 0^-} \frac{f(x) - f(0)}{x - 0}\]
\[ = \lim_{h \to 0} \frac{f(0 - h) - f(0)}{0 - h - 0} \]
\[ = \lim_{h \to 0} \frac{- (0 - h) - 0}{- h}\]
\[ = - 1\]

(RHD at x = 0)

\[\lim_{x \to 0^+} \frac{f(x) - f(0)}{x - 0} \]
\[ = \lim_{h \to 0} \frac{f(0 + h) - f(0)}{0 + h - 0} \]
\[ = \lim_{h \to 0} \frac{0 + h - 0}{h} \]
\[ = 1\]

Thus,  

\[\lim_{x \to 0^-} f(x)\] ≠ 

Hence 

\[f(x)\]  is not differentiable at 

\[x = 0\]