Let a and b be extended real numbers with aa of fx and lim x

Let a and b be extended real numbers with a<b. Prove that if f is a bounded monotone function on the interval (a,b), then lim x-->a+ of f(x) and lim x-->b- of f(x) both exist and are finite.

Solution

If f : A R and c A is an accumulation point of A, then f is continuous at c if and only if limn f(xn) = f(c) for every sequence (xn) in A such that xn c as n . In particular, f is discontinuous at c A if there is sequence (xn) in the domain A of f such that xn c but f(xn) 6 f(c). Let’s consider some examples of continuous and discontinuous functions to illustrate the definition.