Prove if $f$ is continuous at $x_0$ and $f(x_0)>\mu$, then $f(x)>\mu ,\forall x$ in some neighborhood of $x_0$ 
Prove if $f$ is continuous at $x_0$ and $f(x_0)>\mu$, then $f(x)>\mu ,\forall x$ in some neighborhood of $x_0$

My attempt:
$f$ is continuous implies:
$$\forall \epsilon>0, \exists \delta >0 \text{ s.t.} |f(x)-f(x_0)|<\epsilon \forall |x-x_0|<\delta$$
$$\epsilon \geq |f(x)-f(x_0)|=|f(x_0)-f(x)|\geq |f(x_0)|-|f(x)|\geq \mu-|f(x)|$$
i am not sure how to proceed?

Hint:
Set $\epsilon = \frac{|f(x_0)-\mu|}{2}$
