Let

$\displaystyle

f(x) = \left\{

\begin{array}{ll}

x^2 & : x \geq 0\\

x & : x < 0

\end{array}\right.

$

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

Since function f is continuous at a if

$\displaystyle

f(a) = \lim_{x->a} f(x)

$

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

$\displaystyle \lim_{x -> 0} f(x) = 0$

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

$\displaystyle a < 0: f(a) = a = \lim_{x -> a} x$

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.