Question

I am not familiar with "increasing at a point". I know that `f(x) = x^3` is increasing on the whole real line, but since `f'(0)=0` do some people say that it is not increasing at `x = 0`?

Answer 1

There is potential for confusion with this term.

Explanation:

"increasing at a point" could refer to "monotonicity", in which case we would define "increasing at a point" as follows:

`f` is increasing at a point `x = c` if and only if `EE delta > 0 :`

`f(x) < f(c) AA x in (c - delta, c)`

and

`f(x) > f(c) AA x in (c, c + delta)`

The function `f(x) = x^3` passes this test at `x=0`.

On the other hand, a "stationary point" of a function `f` is a point where its derivative is `0`, that is:

`lim_(x->c) (f(x)-f(c))/(x-c) = 0`

So we might prefer to define "increasing at a point" as having a strictly positive derivative at that point.

If we use the "montonicity" definition, then we are left with the counter-intuitive situation that `f(x) = x^3` is "stationary" but "increasing" at `x=0`.

I think it is not a well defined term.