# Show that for any $ε>0$, there are uncountably many points $x\in[a,b]$ such that $f′(x)\in(M−ε,M+ε)$

Let $$f: [a,b]\to R$$ be a continuous function differentiable on $$(a,b)$$.

Define $$M:=\frac{f(b)-f(a)}{b-a}$$.

Show that for any $$ε>0$$, there are uncountably many points $$x\in[a,b]$$ such that $$f′(x)\in(M−ε,M+ε)$$

I know that by MVT, there exists $$x^*\in(a,b)$$ such that $$f'(x^*)=M$$. But I am stuck here. How should I proceed?

PS: this is a homework problem

• Do you know that derivatives have the IVP? Apr 26 '20 at 20:47
• Yes, but doesn't it require the function to be differentiable on [a,b] instead of (a,b)
– user779519
Apr 26 '20 at 20:49
• @ez709, this is an assignment question. Plz note it in your question otherwise honor code is violated. Apr 26 '20 at 22:18

If $$f$$ is linear on $$[a,b]$$, then $$f'(x)=M$$ for every point $$\in(a,b)$$ and we are done. So assume there is $$x_0\in(a,b)$$ with $$\frac{f(x_0)-f(a)}{x_0-a}\ne M$$. Then from continuity of $$x\mapsto \frac{f(x)-f(a)}{x-a}$$ on $$[x_0,b]$$ and the Intermediate Value Theorme, we conclude that for every $$M'$$ between $$M$$ and $$\frac{f(x_0)-f(a)}{x_0-a}$$ (in particular, either for every $$M'\in(M,M+\epsilon)$$ or for every $$M\in(M-\epsilon,M)$$), there exists $$x\in (x_0,b)$$ with $$\frac{f(x)-f(a)}{x-a}=M'$$. Then by the Mean Value Theorme, there exists $$\xi\in (a,x)$$ with $$f'(\xi)=M'$$.

• I am not getting the "from countinuity of x" part. Could you explain more? thx
– user779519
Apr 26 '20 at 21:06

hint

$$f$$ is differentiable at $$(a,b)$$, thus

the set $$E= f'((a,b))$$ is an intervall.

as you said, $$M\in E$$ by MVT.

then M is not an isolated point.