Understanding Limits of Differentiability

[Reckoner]

Can someone explain to me the above? Like how did f(h) become -1?

For all $x < 0,$ $f(x) = -1.$ Since we are taking the left-handed limit ($h$ is approaching 0 from the left), we are dealing with $h < 0.$ Similarly, for the right-handed limit we are looking at where $h > 0$ so that $f(h) = 1.$

 

[Deveno]

One small observation:

The quantity:

\(\displaystyle \frac{(x+h) - f(x)}{h}\)

for \(\displaystyle f(x) = \frac{x}{|x|}\)

has NO MEANING at the point \(\displaystyle x = 0\) because f is UNDEFINED there.

In vague analytical terms, when we take the limit:

\(\displaystyle \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}\)

we require three things:

1) f(a) is defined
2) f(x) is defined for some (possibly very small) open interval (a-|h|,a+|h|)
3) this limit exists

If one looks closely one sees that (1) and (2) together "almost" imply continuity, the limit condition (3) means we don't have some strange "skipping/jumping" behavior near the point a (in fact it says something STRONGER, it is a "smoothness" requirement)

The derivative f'(a) is a certain limit involving expressions with f. I want to point out that f'(a) is a (real) NUMBER, the FUNCTION f' (which has the value f'(a) at the point a, if it exists) is another matter entirely. For the FUNCTION to exist, we require differentiability at many,many points, and not all functions qualify (for example, the function f(x) = |x| has a "corner" at x = 0, and there exist extremely "jagged" functions, which are continuous everywhere but have no derivative anywhere!).

SO differentiabilty of a function (everywhere) is a rather SPECIAL quality, just as continuity is a special quality. In the grand scheme of things, MOST functions are not this special. In a way, calculus misleads us into thinking most functions SHOULD be continuous, or differentiable (because most of the ones we think of trying first, just so happen to qualify). The sad truth is this: the "bizarre" functions are the vast majority, the "nice" functions are a tiny sliver of these, which are the ones we concentrate on, just BECAUSE they are so EASY to work with.