Example 4

The absolute value function y = |x| is continuous but not differentiable at the point x = 0. (See Figure 3.4.4(b).)

We have already shown that the derivative does not exist at x = 0. To see that the function is continuous, we note that for any infinitesimal Δx,

Δy = |0 + Δx| - |0| = |Δx|

and thus Δy is infinitesimal.

Figure 3.4.4