Show that f(x) = |x - 5| is continuous but not differentiable at …

Question

Show that f(x) = |x - 5| is continuous but not differentiable at x = 5.

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Answer

A function f is a differentiable function if and only if $\text{Lf (c)}=\lim\limits_{\text{h}\rightarrow0}\frac{\text{f(a}-\text{h})-\text{f}(\text{a})}{-\text{h}}$ and $\text{Rf}'(\text{c})=\lim\limits_{\text{h}\rightarrow0}\frac{\text{f}(\text{a}-\text{h})-\text{f(a)}}{\text{h}}$ are equal.
Consider, f(x) = |x - 5|
$\therefore\ \text{f(x)}=\begin{cases}-(\text{x}-5),&\text{if x}<5\\\text{x}-5,&\text{if x}\geq5\end{cases}$
For continuity at x = 5,
$\text{L.H.L}=\lim\limits_{\text{x}\rightarrow5^-}(-\text{x}+5)$
$=\lim\limits_{\text{h}\rightarrow0}\big[-(5-\text{h})+5\big]=\lim\limits_{\text{h}\rightarrow0}\text{h}=0$
$\text{R.H.L}=\lim\limits_{\text{h}\rightarrow5^+}(\text{x}-5)$
$=\lim\limits_{\text{h}\rightarrow0}(5+\text{h}-5)=\lim\limits_{\text{h}\rightarrow0}\text{h}=0$
$\text{f}(5)=5-5=0$
$\Rightarrow\ \text{L.H.L}=\text{R.H.L}=\text{f}(5)$
Hence, f(x) is continuous at x = 5
Now, $\text{Lf}'(5)=\lim\limits_{\text{x}\rightarrow5^-}\frac{\text{f(x)}-\text{f}(5)}{\text{x}-5}$
$=\lim\limits_{\text{x}\rightarrow5^-}\frac{-\text{x}+5-0}{\text{x}-5}=-1$
$\text{Rf}'(5)=\lim\limits_{\text{x}\rightarrow5^+}\frac{\text{f(x)}-\text{f}(5)}{\text{x}-5}$
$=\lim\limits_{\text{x}\rightarrow5^+}\frac{\text{x}-5-0}{\text{x}-5}=1$
Lf'(5) ≠ Rf'(5)
So, f(x) = |x - 5| is not differentiable at x = 5.

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