/sci/ - Science & Math

How do I prove that a function is continuous in [math]\mathbb{R}[/math]? I know how to prove that a function is continuous at some point, but I'm not sure how to prove that it's continuous everywhere.
My current approach wrt. piecewise functions was to just examine the point at which the function changes and handwave the rest as "it's composed of elementary functions, so it's continuous ". Is this correct?
Picrel is a function. I just tried to show that for some values of a,b,c it's continuous at -1 and 1 (I came up with [math]b = 0, c = 1, a \in \mathbb{R}[/math] and [math]b = \sqrt{1 + a^2} , c = -a, a \in \mathbb{R}[/math])