What's the difference between $\alpha$-sublevel set of a function f and it's epigraph? 
$\alpha$-sublevel set of f:
$C_\alpha=\{x \in \mathbf{dom} f|f(x)\leq\alpha\}.$
epigraph of f:${\mbox{epi}}f=\{(x,\mu )\,:\,x\in {\mathbb  {R}}^{n},\,\mu \in {\mathbb  {R}},\,\mu \geq f(x)\}\subseteq {\mathbb  {R}}^{{n+1}}.$

I'm not sure what the difference is. It seems like both refers to the domain of f where f(x) lies under a certain number, but with the epigraph that number isn't fixed.
 A: Consider $f(x) = x$, where $\text{dom}(f) = \mathbb R$. As an example, let $\alpha = 1$, so that $C_\alpha = C_1 = (-\infty, 1]$. If we select a point in $C_1$, then we're selecting a number $x$ in the interval $(-\infty, 1]$. 
On the other hand, $\text{epi}(f) = \{(x,\mu): \; x \in \mathbb R, \; \mu \in \mathbb R, \; \mu \geq x\}$. If we select a point in $\text{epi}(f)$, then we're selecting an entire coordinate pair $(x, \mu)$, not just a number. This coordinate lies in the two-dimensional region above the line $f(x) = x$. 
So, for this $f$, a point in $C_\alpha$ is always a number $x$ (for any $\alpha$), but a point in $\text{epi}(f)$ would be an entire coordinate pair $(x, \mu)$. Also, notice that $\alpha$ parametrizes the sublevel sets of $f$, but no such parameter is needed in defining the epigraph of $f$.
(By the way, it would be more precise to write $C_\alpha(f)$ or something similar, just as we write $\text{epi}(f)$, to clarify that the sublevel set is defined relative to a function $f$, but I think this is already understood from context.) 
