I'm trying to prove that if a function is continuous on [a,b] and smooth on (a,b) then there's a point x in (a,b) where f'(x) exists. The definition of smoothness is: f is smooth at x iff [itex] \lim_{h \rightarrow 0} \frac{ f(x+h) + f(x-h) - 2f(x) }{ h } = 0 [/itex].(adsbygoogle = window.adsbygoogle || []).push({});

I'm starting with the simpler case where f(a) = f(b). I know that f has a max x and a min y on [a,b], and that if both occur at a or b, then f is a constant function. So I'm assuming wlog that the max x occurs somewhere in (a,b).

So at x,

[itex] - \frac{ (f(x) - f(x+h)) } {h} \leq 0 [/itex] and

[itex] \frac{f(x-h) - f(x)}{h} \leq 0 [/itex].

This is where I'm getting stuck. I'd like to show that the limits of the one sided difference quotients as [itex] h \rightarrow 0- [/itex] exist and then apply smoothness to show that these must be equal, hence f is differentiable at x. However, I haven't been able to show that they exist - taking the infima of both sides doesn't seem to yield anything, so I'm guessing I might be missign something else. Thanks in advance.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Continuous and smooth on a compact set implies differentiability at a point

**Physics Forums | Science Articles, Homework Help, Discussion**