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## Homework Statement

Proof that, If [itex]f[/itex] is a function such that

(1) [itex]f[/itex] is differentiable at (open) the interval [itex]D[/itex],

(2) [itex]D[/itex] includes [itex]0[/itex] and [itex]f(0)=0[/itex], and

(3) for all [itex]x[/itex] in [itex]D[/itex] other than [itex]0[/itex], [itex]f(x)[/itex] and [itex]x[/itex] have opposite signs

Then

[itex]f'(0)\leq0[/itex]

## Homework Equations

None.

## The Attempt at a Solution

I managed to prove that for all [itex]x[/itex] in [itex]D[/itex] other than [itex]0[/itex]

[itex]\frac{f(x)-f(0)}{x-0}\leq0[/itex]

I don't know how to get from there to the fact that

[itex]lim _{x\rightarrow0} \frac{f(x)-f(0)}{x-0}\leq0[/itex]Any help would be very appreciated. Thanks.

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