There's lots of questions on here about finite differences. In particular, picking the 'best' h value. But what if I want to find the biggest 'h' which bounds to a given tolerance?
On first glance, I would think for a simple two-point forward difference:
$$ f(x+h) - f(x)\over h\\ $$
That the inequality could simply be:
$$ -h/2 \times f''(c) > some-tolerance $$
In this case, my function is simply $$ sin(2x) $$
However, it could easily be any function.
However, after some working out it seems to depend on $c$... which depends on $h$. Where should I go?
If the second derivative is bounded on the interval you want, just use its maximum.