Prove that f is continuous

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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.



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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?
 

Answers and Replies

  • #2
HallsofIvy
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Yes, it is.
 

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