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4. Assume that a continuous function f : R –→ R is T-periodic; that is, f(x + T) = f(x) for all x E R. Prove that for every natural number n there exists Xn E [0, T] such that f(xn) = f(xn +). TT:

4. Assume that a continuous function f : R –→ R is T-periodic; that is, f(x + T) = f(x) for all x E R. Prove that for every natural number n there exists Xn E [0, T] such that f(xn) = f(xn +). TT:

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.5: The Kernel And Range Of A Linear Transformation
Problem 30EQ
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4. Assume that a continuous function f : R → R is T-periodic; that is, f(x + T) = f(x) for all x E R. Prove that for every natural number n there exists Xn E [0, T] such that f(xn) = f(xn +). Hint: Think about the function g(x) = f(x+ ?) – f(x) and the expression n st0) + () +-+ (") T. +...+g T(n – 1) n n
Transcribed Image Text:4. Assume that a continuous function f : R → R is T-periodic; that is, f(x + T) = f(x) for all x E R. Prove that for every natural number n there exists Xn E [0, T] such that f(xn) = f(xn +). Hint: Think about the function g(x) = f(x+ ?) – f(x) and the expression n st0) + () +-+ (") T. +...+g T(n – 1) n n
Expert Solution
Step 1

given a continuous function f:ℝ→ℝ is T-periodic,

defined as

fx+T=fx ∀x∈ℝ

to prove- for every natural number n there exist xn∈0,T

such that 

fxn=fxn+Tn

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