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Question
Give an example of a function which is continuos but not differentiable at at a point.
Solution
Consider a function,
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
Thus,
Hence,
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,
Hence
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