Let
<br />
f(x) = \left\{<br />
\begin{array}{ll}<br />
x^2 & : x \geq 0\\<br />
x & : x < 0<br />
\end{array}\right.<br />

Suppose you want to show f is a continuous function for all x.

Since function f is continuous at a if
<br />
f(a) = \lim_{x->a} f(x)<br />

and a function is continuous on its domain if the above holds for all x in the domain, one could say:

\lim_{x -> 0} f(x) = 0
a \geq 0: f(a) = a^2 = \lim_{x -> a} x^2
a < 0: f(a) = a = \lim_{x -> a} x

Therefore the function f is continuous.

My professor scratched his head for moment before agreeing it made sense, so I'm guessing there's a more conventional form for expressing this idea.
What is it, I can't wait for my next class.