$ f(r) = f( r + \frac{1}{2} ) $

Suppose that $f: [0,1] \rightarrow [0,1]$ is a continuous function satisfying $f(0) = f(1)$ . Show that there is a real number $r$ such that $f(r) = f( r + \frac{1}{2} )$ .

This is a list of Calculus proof based problems that I like. Please avoid posting complete solutions, so that others can work on it.

#Calculus #Proofs

Easy Math Editor

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Comments

On $[0,1]$ define $g(x)=f(x) - f(x+1/2)$ . Note that : $g(0)=-g(1/2)$ since : $f(0)=f(1)$ .

This means that there is $r\in [0,1/2]$ such that : $f(r)=f(r+1/2)$ .

Haroun Meghaichi - 7 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

\( f(r) = f( r + \frac{1}{2} ) \)

Comments