derivatives

Apr 2010
17
0
I can not figure this problem out. Please somebody help me. let f: (a,b)\(\displaystyle \rightarrow\)R, where a,b \(\displaystyle \in\)R be a function and let x \(\displaystyle \in\)(a,b). Prove that if limh\(\displaystyle \rightarrow\)0|f(x+h)-f(x)|=0, then lim h\(\displaystyle \rightarrow\)0|f(x+h)-f(x-h)| = 0.
 
Oct 2007
722
168
I believe it's along the lines of...

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

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

\(\displaystyle \leq \lim_{h \to 0} |f(x+h) - f(x)| + \lim_{h \to 0} |f(x) - f(x-h)| \to 0+0 = 0 \textrm{ as } h \to 0\) 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...