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Say real functions g(x) and p(y) are continuous and f(x,y) = g(x)p(y). How to proof rigorously the continuity of f in a point (x1,y1)?

In other words, how to obtain l g(x)p(y) - g(x1)p(y1) l < epsilon (for any epsilon).

I can prove that l g(x)p(y1) - g(x1)p(y) l < any epsilon, but I cant see how to go from here to there. I am trying all variations of the triangular inequality, to no avail.

Thanks for your help.

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# Proving continuity of f(x,y) = g(x)p(y)

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