# Prove that the function $$f$$ given by $f(x) = | x - 1 |, x \in R\;$ is not differentiable at $\; x = 1$.

Toolbox:
• The given function may be written as $f(x)=\left \{\begin{array}{1 1}x-1 &if\;x\geq 1\\1-x &if\;x<1\end{array}\right.$
• If $\lim_{h \to 0}\large\frac{f(c+h)-f(c)}{h}$ does not exist.
• We say that f is not differentiable at c.
Step 1:
RHD=Right hand derivative at $x=1$
$\lim_{h \to 0}\large\frac{f(1+h)-f(1)}{h}$
$\Rightarrow \lim_{h \to 0}\large\frac{[(1+h)-1]-(1-1)}{h}$
$\Rightarrow\lim_{h \to 0}\large\frac{h}{h}$=1
Step 2:
LHD=Left hand derivative at $x=1.$
$\lim_{h \to 0}\large\frac{f(1+h)-f(1)}{-h}$
$\Rightarrow \lim_{h \to 0}\large\frac{[1-(1-h)]-(1-1)}{-h}$
$\Rightarrow\lim_{h \to 0}\large\frac{h}{-h}$= -1
Step 3:
LHD= -1
RHD=1
LHD$\neq$ RHD.
Hence f is not differentiable at $x=1$
edited May 8, 2013