If $f$ is continuous at $x\_0$ and $f(x\_0)>M$, then $f(x)>M$ for all $x$ some neighborhood of $x\_0$. 

My attempt is below. 

From the assumptions above, we have that $f(x\_0) > M = f(x\_1)$ for some $x\_1$ that $f$ is defined. Suppose that $x\_0 > x\_1$, then take $\delta >0$ such that $x\_1\not\in (x\_0-\delta,x\_0+\delta)$, which gives $f(x) >f(x\_1) = M\ \forall x\in(x\_0-\delta,x\_0+\delta)$ since $f$ is continuous at $x\_0$. 

Does this hold? I feel like there is a better approach to this. Thanks!