# derivatives

#### janae77

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.

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