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6.3 Examples of non Differentiable Behavior

A function which jumps is not differentiable at the jump nor is one which has a cusp, like | x | has at x = 0 .

Generally the most common forms of non-differentiable behavior involve a function going to infinity at x , or having a jump or cusp at x .

There are however stranger things. The function sin ⁡ ( 1 x ) , for example is singular at x = 0 even though it always lies between -1 and 1. Its hard to say what it does right near 0 but it sure doesn't look like a straight line.

If the function f has the form f = g h , f will usually be singular at argument x if h vanishes there, h ( x ) = 0 . However if g vanishes at x as well, then f will usually be well behaved near x , though strictly speaking it is undefined there.

We usually define f at x under such circumstances to be the ratio of the linear approximation at x to g to that to h very near x , which means we define f ( x ) to be f = g ' ( x ) h ' ( x ) , when, of course the denominator here does not vanish. (If the denominator does vanish and the numerator vanishes as well, you can try to define f ( x ) similarly as the ratio of the derivatives of these derivatives, etc.)

This kind of thing, an isolated point at which a function is not defined, is called a "removable singularity" and the procedure for removing it just discussed is called "l' Hospital's rule" .

An example is sin ⁡ x x at x = 0 .

Continuous but non differentiable functions

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