91Ó°ÊÓ

Suppose that \(f\) is a function defined on the real line with the property that \(f(x+y)=f(x)+f(y)\) for all \(x, y .\) Suppose that \(f\) is continuous at 0. Show that \(f(x)=C x\) for all \(x\) and some number \(C\).

Prove that the function \(f(x)=x^{2}\) is continuous at every point of \(\mathbb{R}\) using the \(\delta-\varepsilon\) form of continuity,

Let \(f\) be a continuous function on an open interval \((a, b)\). Suppose that \(f\) has no local maximum or local minimum at any point. Show that \(f\) must be monotonic.

Prove that the limit \(\lim _{x \rightarrow 0} \sin (1 / x)\) fails to exist by converting to a statement about sequences.

Show that one of the limits \(\lim _{x \rightarrow 0+} f(x)\) and \(\lim _{x \rightarrow 0-} f(x)\) of the function $$ f(x)=e^{1 / x} $$ at 0 is infinite and one is finite. What can you say about the limits $$ \lim _{x \rightarrow \infty} f(x) \text { and } \lim _{x \rightarrow-\infty} f(x) ? $$