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:

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

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 $f : ℝ \to ℝ$ is T-periodic,

defined as

$f (x + T) = f (x) \forall x \in ℝ$

to prove- for every natural number $n$ there exist $x_{n} \in [0, T]$

such that

$f (x_{n}) = f (x_{n} + \frac{T}{n})$

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