Let p be a real number and let f: ℝ →ℝ be a continuous function such that f(x+p)=f(x) for all x

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Let p be a real number and let f: ℝ →ℝ be a continuous...

Let $p$ be a real number and let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $f(x+p)=f(x)$ for all $x \in \mathbb{R}$. (Such a function is said to be periodic.) Show that the integral $\int_{a}^{a+p} f(t) d t$ has the same value for every real number $a$. (Hint: Part (ii) of Proposition 6.21.)

Key Concepts

Periodic Functions

A periodic function is one that repeats its values at regular intervals; that is, there exists a constant period p such that f(x + p) = f(x) for every x in its domain. This property implies that the function’s behavior is completely determined by its values on any interval of length p, making it a fundamental concept in analysis and various applied fields such as signal processing and differential equations.

Translation Invariance of Definite Integrals

The translation invariance property of definite integrals states that if a function is integrated over an interval of a fixed length, then shifting the interval by any amount does not change the value of the integral, provided the function exhibits the same behavior on every such interval. For periodic functions, this invariance is derived from the repeating nature of the function, ensuring that the integral over any interval of length equal to the period is identical.

Substitution in Integration

Substitution is a powerful technique in integration that allows one to change the variable of integration to simplify the integral or to shift the limits. In the context of periodic functions, substituting variables can clearly demonstrate that the integral from any point a to a + p is equivalent to the integral over one period, since the periodicity ensures the function’s values repeat across any interval of length p.

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