Let
g: R → R and f: R → R. Assume that f(y) is uniformly continuous
on the set R. Then for any ε > 0 there exists a δ > 0 such
that |y' – y''| < δ implies |f(y') – f(y'')| < ε. If
g(x) is uniformly continuous on the set R, then there exists a η >
0 such that |x' – x''| < η implies |g(x') – g(x'')| < δ.

From
the above it follows that for any ε > 0 there exists a η > 0
such that |x' – x''| < η implies

|f(g(x'))
– f(g(x''))| < ε. So, the composition f(g(x)) is shown to be
uniformly continuous on the set R.

Apr 10th, 2015

Are you studying on the go? Check out our FREE app and post questions on the fly!