Answer to Question #284348 in Real Analysis for Reens

Question

Are the statement true or false?give reason for your answers: the function,f(x) =sin^2x is uniformly continuous in the interval [0,π]

Answer 1

Since the function $f(x)=\sin^2x$ is a composition of the elementary continuous function $g(x)=x^2$ and $h(x)=\sin x,$ we conclude that the function $f$ is continuous in the interval $[0,π].$

The Heine-Cantor theorem asserts that every continuous function on a compact set is uniformly continuous. In particular, if a function is continuous on a closed bounded interval of the real line, it is uniformly continuous on that interval. It follows that the function $f(x)=\sin^2x$ is uniformly continuous in the interval $[0,π].$

Answer: true