- #1
DJBruce
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Homework Statement
"A real valued function, f, has the following property:
[tex]\left|f\right|[/tex] is differentiable at [tex] x=0 [/tex]
Prove that if we specify that f is continuous at 0, then f is also differentiable at 0."
Homework Equations
Since [tex]\left|f \right|[/tex] is differentiable we know the following:
[tex]\lim_{x\to0} \frac{\left|f(x)\right|-\left|f(0)\right|}{x} [/tex] exists, which means:
[tex]\vee \epsilon>0 \exists \delta >0 [/tex] st if [tex]\left|x\right|<\delta, \left|\frac{\left|f(x)\right|-\left|f(0)\right|}{x} - \left|f'(0)\right|\right|<\epsilon[/tex]
Likewise, since f is continuous at 0 we know:
[tex]\vee \epsilon>0 \exists \delta >0 [/tex] st if [tex] \left|x\right|<\delta, \left|f(x) - f(0)\right|<\epsilon[/tex]
We want to show:
[tex]\lim_{x\to0} \frac{f(x)-f(0)}{x} [/tex] exists, which mean we want to show:
[tex]\vee \epsilon>0 \exists \delta >0[/tex] st if [tex] \left|x\right|<\delta, \left|\frac{f(x)-f(0)}{x} - f'(0)\right|<\epsilon[/tex]
The Attempt at a Solution
I am not really sure where to go. I thought of using triangle inequality, but that would won't work since it implies the other direction.