Which of the following statements are true?

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(B, D) For a periodic function $f(x)$ with period $T$,the integral over any interval of length $T$ is constant,i.e.,$\int_a^{a+T} f(x) dx = \int_0^T f

Vedclass · Accepted Answer

$f$ defined in $(C)$ is periodic for all $T \in \mathbb{Q} \setminus \{0\}$.

Vedclass · Answer

(B, D) For a periodic function $f(x)$ with period $T$,the integral over any interval of length $T$ is constant,i.e.,$\int_a^{a+T} f(x) dx = \int_0^T f(x) dx$. Thus,statement $(B)$ is true and $(A)$ is false.For the Dirichlet function $f(x) = \begin{cases} 1, & x \in \mathbb{Q} \ 0, & x 
otin \mathbb{Q} \end{cases}$,$f(x+T) = f(x)$ holds for any rational $T$ because if $x$ is rational,$x+T$ is rational,and if $x$ is irrational,$x+T$ is irrational. Thus,$f$ is periodic for all rational $T$. Statement $(D)$ is true.

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