# Prove that f is continuous

1. Oct 3, 2012

### drawar

1. The problem statement, all variables and given/known data
Suppose that f is an odd function satisfying $\mathop {\lim }\limits_{x \to {0^ + }} f(x) = f(0)$. Prove that f(0)=0 and f is continuous at x=0.

2. Relevant equations

3. The attempt at a solution
Since f is an odd function $$f(0) = - f(0) \Rightarrow f(0) = 0$$
Let t=-x, then when $x \to {0^ + },t \to {0^ - }$
$\mathop {\lim }\limits_{x \to {0^ + }} f(x) = \mathop {\lim }\limits_{t \to {0^ - }} f( - t) = - \mathop {\lim }\limits_{t \to {0^ - }} f(t) = - \mathop {\lim }\limits_{x \to {0^ - }} f(x) = f(0) = 0$
Therefore $\mathop {\lim }\limits_{x \to {0^ + }} f(x) = \mathop {\lim }\limits_{x \to {0^ - }} f(x) = f(0)$, which implies that f(x) is continuous at 0.

Is my working correct?

2. Oct 3, 2012

### HallsofIvy

Staff Emeritus
Yes, it is.
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