Basic Analysis I — Real Analysis · Problem 3.6.6

Continuous Functions — Jiri Lebl

Mathematics — Real Analysis · monotone-functions-continuity

Problem

Suppose

S \subset \mathbb{R}

, and

f : S \to \mathbb{R}

is an increasing function. Prove: a) If

c

is a cluster point of

S \cap (c, \infty)

, then

\displaystyle\lim_{x \to c^+} f(x) < \infty

. b) If

c

is a cluster point of

S \cap (-\infty, c)

and

\displaystyle\lim_{x \to c^-} f(x) = \infty

, then

S \subset (-\infty, c)