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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)?

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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, $f(U)\subset [0,\epsilon)$.

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