real analysis question

label Mathematics
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Apr 10th, 2015

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

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