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#?

1 Answer
Nov 23, 2015

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.