Differentiable, continuous function... "show that"

**MMath09** · May 19th 2009, 05:40 AM

If $f$ is continuous on $[0,1]$ and differentiable at a point $x \in [0,1]$ , show that, for some pair $m,n \in \mathbb{N}$ ,
$\left | \frac{f(t)-f(x)}{t-x}\right | \leq n$ whenever $0 \leq |t-x| \leq \frac{1}{m}$

Since it's differentiable at x I know $\lim_{t \to x}\frac{f(t)-f(x)}{t-x}$ exists...
And it's continuous so I know for that for any $\epsilon > 0$ there's a $\delta > 0$ so that $|f(t)-f(x)| < \epsilon$ when $|t-x| < \delta$

Not sure how to put it all together...

**HallsofIvy** · May 19th 2009, 06:29 AM

Try using the mean value theorem instead.

**MMath09** · May 19th 2009, 07:21 PM

Originally Posted by HallsofIvy

Try using the mean value theorem instead.

Wouldn't that require the function to be differentiable on (0,1) ? I only have differentiability at a single point in [0,1]...

I don't follow :S

**HallsofIvy** · May 20th 2009, 12:55 PM

Oops, right, I didn't see that! Sorry. I guess you were right to use the definition of "differentiable at a point".

Let n be any integer strictly larger than f'(x)+1. Certainly there exist [itex]\delta> 0[/itex] such that $\frac{|f(t)- f(x)||}{|t-x|}< f'(x)+ 1< n$ And, since $\frac{1}{\delta}$ is a real number, there exist [tex]m> \frac{1}{\delta}[/itex]. If $|t- x|< \frac{1}{m}< \delta$ then |\frac{f(t)- f(x)}{x-t}|< n.

**MMath09** · May 21st 2009, 03:01 AM

Originally Posted by HallsofIvy

Certainly there exist $\delta> 0$ such that $\frac{|f(t)- f(x)|}{|t-x|}< f'(x)+ 1< n$

when |t-x| < delta?

I don't get this bit

Thanks for your help, I'm almost there

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