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.