Another question, sorry Final Friday, senior year, this class is horrible

Let f: D-->R be uniformly continuous on D and let cED. Prove using an

epsilon-delta argument cf is uniformly continuous on D.

R = Real Numbers

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- May 6th 2008, 08:20 PMVitava61More Real Analysis: Epsilon-Delta: Uniformly Continuous
Another question, sorry Final Friday, senior year, this class is horrible

Let f: D-->R be uniformly continuous on D and let cED. Prove using an

epsilon-delta argument cf is uniformly continuous on D.

R = Real Numbers - May 6th 2008, 09:45 PMTheEmptySet
let $\displaystyle \epsilon > 0$

Since f is uniformly continuous there exisits a delta for every epsilon such that

$\displaystyle |x-y|< \delta$ $\displaystyle |f(x)-f(y)|<\frac{\epsilon}{|c|}$

Now for $\displaystyle |x-c|< \delta$

$\displaystyle |cf(x)-cf(y)|=|c||f(x)-f(y)|<|c|\cdot \frac{\epsilon}{|c|}=\epsilon$

QED