# Pointwise continuity of dominated non-negative function

Let $$M$$ be a topological space and $$f, g : M\rightarrow\mathbb{R}_+$$ be two non-negative functions with $$f(x_0)=g(x_0)=0$$ for some $$x_0\in M$$.

Suppose further that $$f(x)\leq g(x)$$ for each $$x\in M$$, and that $$g$$ is continuous at $$x_0$$.

I was wondering if this is sufficient to conclude that $$f$$ must also be continuous at $$x_0$$?

Clearly this holds if $$M$$ is a metric space, but is it also true for general topological spaces (the topology on $$\mathbb{R}_+$$ shall always be the Euclidean (subspace) topology)?

First recall the definition of continuity. On a topological space $$M$$, a function $$f$$ is continuous at $$x_0$$ if for every neighborhood $$V$$ of $$f(x_0)$$, there is a neighborhood $$U$$ of $$x_0$$ so that $$f(U)\subset V$$. Now, by assumption, $$g$$ is continuous at $$x_0$$, so let $$\epsilon > 0$$, and find a neighborhood $$U$$ of $$x_0$$ such that $$g(U)\subset[0,\epsilon)$$. Since $$f\le g$$ everywhere, $$f(U)\subset [0,\epsilon)$$.