Thread: How to prove $f(a)\leq f(b)$?

Dear all, suppose is a continous function on [TEX[a,b][/TEX]. is countable. Assume is differentiable on , the the derivative . Show that

Note: We assume only that is differentiable on

I do not know how to prove. Do you know? Help me, thank you...

Since E is a countable subset of (a,b), let E = {e_i} with a < e₁ < e₂< ... < b.

Then we have that f is continuos on [a, e₁], and differentiable on (a, e₁). The mean value theorem applies and we find that f(a) f(e₁). We can prove this by contradiction. Assuming f(a) > f(e₁), we would have by the MVT a c₁, s.t. f'(c₁) = (f(e₁​)-f(a))/(e₁-a) < 0, a contradiction to f'(x) 0.

Similiarly, we can find that f(e₁) f(e₂). Continuing in this way we can argue that for any e_i < e_j, f(e_i) f(e_j).

I think this may be a step toward proving the desired, f(a) f(b), but I haven't yet worked it completely. Will finish and post the rest soon, unless you or someone else can finish it.

Do anyone knows how to prove it? Thank you ...