CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question
Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY3 Marks
Question
Give an example of a function which is continuos but not differentiable at at a point.
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Answer
Consider a function, $\text{f(x)}=\begin{cases}\text{x}, & \text{x}> 0\\-\text{x}, & \text{x}\leq 0\end{cases}$ This mod function is continuous at x = 0 but not differentiable at x = 0. Continuity at x - 0, We have: (LHL at x = 0) $\lim_\limits{\text{x}\rightarrow0^{-}}\text{f(x)}$ $=\lim_\limits{\text{x}\rightarrow0}\text{f}(0-\text{h})$ $=\lim_\limits{\text{x}\rightarrow0}-(0-\text{h})$ $=0$ (RHL at x = 0) $\lim_\limits{\text{x}\rightarrow0^{+}}\text{f(x)}$ $=\lim_\limits{\text{x}\rightarrow0}\text{f}(0+\text{h})$ $=\lim_\limits{\text{x}\rightarrow0}(0+\text{h})$ $=0$ and f(0) = 0 Thus, $\lim_\limits{\text{x}\rightarrow0^{-}}\text{f(x)}=\lim_\limits{\text{x}\rightarrow0^{+}}\text{f(x)}=\text{f}(0).$ Hence, f(x) is continuous at x = 0. Now, we will check the differentiability at x = 0, we have: (LHL at x = 0) $\lim_\limits{\text{x}\rightarrow0^{-}}\frac{\text{f(x)}-\text{f}(0)}{\text{x}-0}$ $=\lim_\limits{\text{h}\rightarrow0}\frac{\text{f}(0-\text{h})-\text{f}(0)}{0-\text{h}-0}$ $=\lim_\limits{\text{h}\rightarrow0}\frac{-(0-\text{h})-0}{-\text{h}}=-1$ (RHL at x = 0) $\lim_\limits{\text{x}\rightarrow0^{+}}\frac{\text{f(x)}-\text{f}(0)}{\text{x}-0}$ $=\lim_\limits{\text{h}\rightarrow0}\frac{\text{f}(0+\text{h})-\text{f}(0)}{0+\text{h}-0}$ $=\lim_\limits{\text{h}\rightarrow0}\frac{(0+\text{h})-0}{-\text{h}}=1$ Thus, $\lim_\limits{\text{h}\rightarrow0^{-}}\text{f(x)}\neq\lim_\limits{\text{h}\rightarrow0^{+}}\text{f(x)}$ Hence f(x) is not differentiable at x = 0.
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