Hi,

I do not know how to solve below problem; maybe someone can guide me through it:

Prove that is continuous on for any positive integer .

Thank you.

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- Aug 29th 2013, 08:12 AMGiboReal analysis - continuity of a function
Hi,

I do not know how to solve below problem; maybe someone can guide me through it:

Prove that is continuous on for any positive integer .

Thank you. - Aug 29th 2013, 08:48 AMemakarovRe: Real analysis - continuity of a function
It's easy to show that is not only continuous, but uniformly continuous on every closed segment . Indeed, fix a and assume . We have

(this is checked by multiplying the right-hand side). The second factor in the right-hand side has terms and . Therefore, . The expression is a constant for each fixed , so a small variation in produces only a small variation in . It's easy to turn this into an proof. - Aug 29th 2013, 10:10 AMHartlwRe: Real analysis - continuity of a function
It also follows from f(x)g(x) is conntinuous if f(x) and g(x) are.

- Aug 31st 2013, 12:02 PMGiboRe: Real analysis - continuity of a function
Thank you very much for explaining that. Helped a lot.