Question

How to find points where a function us not differentiable?

Answer 1

To understand where a function’s derivative does not exist, we need to recall what normally happens when a function \displaystyle f\left(x\right)f(x) has a derivative at \displaystyle x=ax=a . Suppose we use a graphing utility to zoom in on \displaystyle x=ax=a . If the function \displaystyle f\left(x\right)f(x) is differentiable, that is, if it is a function that can be differentiated, then the closer one zooms in, the more closely the graph approaches a straight line. This characteristic is called linearity.

Look at the graph in Figure 8. The closer we zoom in on the point, the more linear the curve appears.

We might presume the same thing would happen with any continuous function, but that is not so. The function \displaystyle f\left(x\right)=|x|f(x)=∣x∣, for example, is continuous at \displaystyle x=0x=0, but not differentiable at \displaystyle x=0x=0. As we zoom in close to 0 in Figure 9, the graph does not approach a straight line. No matter how close we zoom in, the graph maintains its sharp corner.We zoom in closer by narrowing the range to produce Figure 10 and continue to observe the same shape. This graph does not appear linear at 
\displaystyle x=0x=0.