Question

Let f: r to r be a continuous function such that lim X tends to + infinity and lim X tends to - infinity exist and both are finite.prove that f is uniformly continuous on r.

Answer 1

Step-by-step explanation:

Take an > 0. Since lims→f(x) v(s) = v(f(x)), there exists 1 > 0 such that if s ∈ I and |s − f(x)| < 1

then |v(s) − v(f(x))| < . Now, f is continuous at x, so there exists δ > 0 such that if t ∈ [a, b] is such

that |t − x| < δ then |f(t) − f(x)| < 1. Since I contains f([a, b]), we automatically have f(t) ∈ I, and also

|f(t) − f(x)| < 1 from above, so |v(f(t)) − v(f(x))| < as desired. This shows that limt→x v(f(t)) = v(f(x)),

i.e. that the composition v ◦ f is continuous at x