Is an Odd Function with a Limit at Zero Always Continuous?

[drawar]

Homework Statement

Suppose that f is an odd function satisfying [itex]\mathop {\lim }\limits_{x \to {0^ + }} f(x) = f(0)[/itex]. Prove that f(0)=0 and f is continuous at x=0.

Homework Equations

The Attempt at a Solution

Since f is an odd function [tex]f(0) = - f(0) \Rightarrow f(0) = 0[/tex]
Let t=-x, then when [itex]x \to {0^ + },t \to {0^ - }[/itex]
[itex]\mathop {\lim }\limits_{x \to {0^ + }} f(x) = \mathop {\lim }\limits_{t \to {0^ - }} f( - t) = - \mathop {\lim }\limits_{t \to {0^ - }} f(t) = - \mathop {\lim }\limits_{x \to {0^ - }} f(x) = f(0) = 0[/itex]
Therefore [itex]\mathop {\lim }\limits_{x \to {0^ + }} f(x) = \mathop {\lim }\limits_{x \to {0^ - }} f(x) = f(0)[/itex], which implies that f(x) is continuous at 0.

Is my working correct?

 

[HallsofIvy]

Yes, it is.