Consider the following two statements about the function f (x) = x: P. f (x) is continuous for all real values of x Q. f (x) is differentiable for all real values of x Which of the following is TRUE?

$\because$        f(x) = |x|

X-X Y0 = $\begin{cases} x, & x \ge 0 \\ ix, & x < 0 \end{cases}$ Graph of f(x) will be -

We can easily see from the above graph that f(x) is continuous for all real values of x. But f(x) is not differentiable for all real values of x eg; at x = 0 So, Let hand derivative = $h \rightarrow0 $$\dfrac{f(0 -h) -f(0)}{-h}$ = $h \rightarrow0 $$\dfrac{-(0 -h)}{-h}$ And Right hand derivative = $h \rightarrow0 $$\dfrac{f(0 -h) -f(0)}{-h}$ = $h \rightarrow0 $$\dfrac{h}{h} = 1$ As if, Left hand derivative $\ne$Right hand derivative. So, f(x) is not differentiable at x = 0 hence it is not differentiable for all real values of x.