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zetafunc.

Guest

Prove Piecewise-Defined Function is C^∞-smooth for all x ∈ R

Hi, not sure what to do here.

Here is a piecewise-defined function h(x);

h(x) = 0 if x ≤ 0
h(x) = e[sup]-1/x[/sup] if x > 0

Prove that h(x) is C[sup]∞[/sup]-smooth over the entire domain of Reals.

For x ≤ 0 I think I have shown this. But I'm not sure what to do for the second 'piece'. Taking nth derivatives of h(x) at x > 0 makes me think there is some kind of pattern but I'm not sure how I can show that it is thus C[sup]∞[/sup]-smooth.

Thanks.

zetafunc.

Guest

Re: Prove Piecewise-Defined Function is C^∞-smooth for all x ∈ R

Wait, h(x) isn't analytic at x = 0...

zetafunc.

Guest

Re: Prove Piecewise-Defined Function is C^∞-smooth for all x ∈ R

bump