91Ó°ÊÓ

A function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is said to be periodic on \(\mathbb{R}\) if there exists a number \(p>0\) such that \(f(x+p)=f(x)\) for all \(x \in \mathbb{R}\). Prove that a continuous periodic function on \(\mathbb{R}\) is bounded and uniformly continuous on \(\mathbb{R}\).

If \(x \in \mathbb{R}, x>0\), and if \(r, s \in \mathbb{Q}\), show that \(x^{r} x^{s}=x^{r+s}=x^{s} x^{r}\) and \(\left(x^{r}\right)^{s}=x^{r s}=\left(x^{s}\right)^{r}\).

Let \(f\) and \(g\) be Lipschitz functions on \(A\). (a) Show that the sum \(f+g\) is also a Lipschitz function on \(A\). (b) Show that if \(f\) and \(g\) are bounded on \(A\), then the product \(f g\) is a Lipschitz function on \(A\). (c) Give an example of a Lipschitz function \(f\) on \([0, \infty)\) such that its square \(f^{2}\) is not a Lipschitz function.

If \(f:[0,1] \rightarrow \mathbb{R}\) is continuous and has only rational [respectively, irrational] values, must \(f\) be constant? Prove your assertion.

Let \(I:=[a, b]\) and let \(f: I \rightarrow \mathbb{R}\) be a (not necessarily continuous) function with the property that for every \(x \in I\), the function \(f\) is bounded on a neighborhood \(V_{\delta_{\lambda}}(x)\) of \(x\) (in the sense of Definition 4.2.1). Prove that \(f\) is bounded on \(I\).