# Showing if something is continuous in Topology

If $f : X \to \mathbb{R}$ is continuous,

I want to show that $(cf)(x) = cf(x)$ is continuous, where $c$ is a constant.

Attempt: If $f$ is continuous, then we want to show that the inverse image of every open set in $\mathbb{R}$ is an open set of $X$. Choose an open interval in $\mathbb{R}$.

Thats as far as I got. :(

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What is it that you're trying to show? Do you want to show that $cf(x)$ is continuous? Or do you want to show that the function notated by $(cf)(x)$ is the same as the function $cf(x)$? – mixedmath Feb 26 at 23:46
I think you mean you want to show $(cf)(x)=cf(x)$ is continuous. – Grumpy Parsnip Feb 26 at 23:46
Sorry about that. Jim, you are right. Thats what I am trying to solve. – user64013 Feb 26 at 23:48
more generally, show that the composition of two continuous functions is continuous. or, in particular here, $c^{-1}U$ is an open interval in $\mathbb{R}$ when $U$ is an open interval in $\mathbb{R}$ – yoyo Feb 26 at 23:55
Let $g:\mathbb{R}\to\mathbb{R}$ be $g(y) = cy$. Your function is $g\circ f$. – Willie Wong Feb 27 at 0:19