1

Let $f: [a,b] \rightarrow \mathbb{R}$ continous on [a,b] and differentiable on $(a,b)$. Asume there exist some $c \in \mathbb{R}$ so $f'(x) \rightarrow c$ for $x \rightarrow a+$. Show $f$ is differentiable at the endpoint with $a$ with $f'(a)=c$

Here's what I'm thinking...

I need to prove $\frac{f(x) - f(a)}{x - a}\rightarrow c$ as $x \rightarrow a+$. My guess would be to use the mean value theorem and replacing $b$ with $x$, so there exists a $\xi$ so $f'(\xi) = \frac{f(x) - f(a)}{x - a}$ where $\xi \rightarrow a$ as $x \rightarrow a+$? (since $\xi \in (a,x)$). But is this valid?

njlieta
  • 411

1 Answers1

3

Ok, so, we want to know if limit $\frac{f(x)-f(a)}{x-a}$ as $x\rightarrow a^+$ exists or not..

As $f$ is cont. $\lim_{x\rightarrow a}f(x)=f(a)$ and so the fraction above would then become $\frac{0}{0}$..

Here comes the Boss... L hos rule...

$\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}=\lim_{x\rightarrow a}\frac{f'(x)-0}{1-0}=\lim_{x\rightarrow a}f'(x)=c$

  • Is it safe to use the limit as $x \rightarrow a$ since $f$ is only differentiable on $(a,b)$? Does $\lim_{x\rightarrow a+}f'(x)=\lim_{x\rightarrow a}f'(x)$ in this case? – njlieta Mar 05 '16 at 11:19
  • 1
    i mean $x\rightarrow a^+$ always –  Mar 05 '16 at 11:23