derivatives

**janae77** · May 26th 2010, 02:49 AM

I can not figure this problem out. Please somebody help me. let f: (a,b) $\rightarrow$ R, where a,b $\in$ R be a function and let x $\in$ (a,b). Prove that if limh $\rightarrow$ 0|f(x+h)-f(x)|=0, then lim h $\rightarrow$ 0|f(x+h)-f(x-h)| = 0.

**Deadstar** · May 26th 2010, 04:43 AM

I believe it's along the lines of...

$\lim_{h \to 0} |f(x+h) - f(x-h)|$

$= \lim_{h \to 0} |f(x+h) - f(x) + f(x) - f(x-h)|$

$\leq \lim_{h \to 0} |f(x+h) - f(x)| + \lim_{h \to 0} |f(x) - f(x-h)| \to 0+0 = 0 \textrm{ as } h \to 0$ where we used the triangle inequality in this step.

You might need to make it a bit more rigorous but I think this is what you have to do...

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