A continuous function which is not differentiable, proof

Set f(x) = |x|.

First let's show it's continuous.

If x > 0, f(x) = x is continuous, if x <0, then f(x) = -x is also continuous. We just need to show it's continuous at x = 0.

Note,

lim x->0^+ |x| = lim x->0^+  x = 0  and lim x > 0^- |x| = lim x->9^- (-x) = 0, whence
lim x->0 |x| = 0 = f(0), so continuity has been shown.

This function is not differentiable at zero. To see this, note,

lim x->0^- (f(x) - f(0))/(x - 0)
lim x->0^- (|x| - 0)/(x - 0)
lim x->0^- |x|/x
lim x->0^- (-x/x)
lim x->0^- (-1)
-1

Likewise,

lim x->0^+ (f(x) - f(0))/(x - 0)
lim x->0^+ (|x| - 0)/(x - 0)
lim x->0^+ |x|/x
lim x->0^+ (x/x)
lim x->0^+ 1
1

Therefore lim x->0 (f(x) - f(0))/(x - 0) DNE, so the function is not differentiable.

This is worth knowing how to do, it might save your life some day!