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Let $(X,\mathcal T)$ be a topological space and the functions: $$f:X\to \Bbb R$$ $$g:X\to \Bbb R^+$$ be continuous. Are the functions $f(x)g(x)$, $f(x)-g(x)$ and $\frac{f(x)}{g(x)}$ continuous?

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    This has bean dealt with before. See here: http://math.stackexchange.com/questions/215109/continuity-of-f-cdot-g-and-f-g-on-standard-topology/308013#308013 – Christian Blatter Mar 12 '13 at 12:41
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    And you don't need positive values for $g$, but nonzero values is enough. – GEdgar Mar 12 '13 at 13:00

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Yes, because all these operations $\Bbb R\times\Bbb R\to\Bbb R$ are continuous, and the mapping $X\to\Bbb R,\ \ x\mapsto (f(x),g(x))$ is continuous, too, and the composition of continuous functions is continuous, too.

Berci
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