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Could anyone help me with the following question?

The periodic function $g$ is defined on $\mathbb{R}$ by $g(x)=f(x)$ for $0\le x<a$ and $g(x)=g(x+a)$ for all $x$, for some $a>1$. It is given that $g$ is a continuous function. Find the exact value of $a$.

Thanks.

user1097856
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1 Answers1

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Hint Assume that $A=\{x>0\mid f(x)=f(0)\}\neq \emptyset$ and suppose that for $a = \min\{x\mid x\in A\}$, $f$ is continuous on $[0,a]$. Show that $a$ is satisfies the required conditions. What happens if $A=\emptyset$? and if $f$ is not continuous on $[0,a]$?

Surb
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  • Thanks for your reply. May i know why u made those assumptions? I don't understand the part of f(x)=f(0). – user1097856 Feb 05 '15 at 13:49
  • @user1097856 suppose that $f(a)\neq f(0)$ then what can you say about the continuity of $g$ around $a$ (or $0$ if you prefer)? Remember that $g$ is periodic. – Surb Feb 05 '15 at 14:09
  • is it because the period of the function is a, so f(a)=f(0)? – user1097856 Feb 06 '15 at 14:00