- #1

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If [tex]f[/tex] is continuous on [tex][a,b][/tex] and differentiable at a point [tex]c \in [a,b][/tex], show that, for some pair [tex]m,n \in \mathbb{N}[/tex],

[tex]\left | \frac{f(x)-f(c)}{x-c}\right | \leq n[/tex] whenever [tex]0 \leq |x-c| \leq \frac{1}{m}[/tex]

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Since it's differentiable at c I know [tex]\lim_{x \to c}\frac{f(x)-f(c)}{x-c}[/tex] exists...

And it's continuous so I know for whatever [tex]\epsilon > 0[/tex] I pick there's a [tex]\delta > 0[/tex] so that [tex]|f(x)-f(c)| < \epsilon[/tex] when [tex]|x-c| < \delta[/tex]

So I guess 1/m could be delta? And then... ? Not sure how to round this one off