find where f is continuous in f(x)= xcotx

Question

determine where f is continuous

Answer 1

When x=πn, where n is an integer, cot(x) is undefined (because cot(x)=cos(x)/sin(x) and sin(0)=0); but x is continuously defined, so f(x) is continuous as long as cot(x) is defined. These are the intervals in which f(x) is continuous:

(-π,0), (0,π), (π,2π), ... (-2π,-π). Note that x=0 is excluded.

This can be generalised: (nπ,(n+1)π).

Note that f(0) is not defined, although the graph looks continuous. xcot(x)=x/tan(x), and when x is very small, tan(x)≈x, so as x→0, f(x)→1. The graph looks continuous at (0,1) but it is not.