Continuity and piecewise defined functions

If a function is defined in pieces, and if the definition changes at $x=a$, then we use the definition for $x \lt a$ to compute $\displaystyle{\lim_{x \to a^-} f(x)}$, we use the definition at $x=a$ to compute $f(a)$, and the definition for $x \gt a$ to compute $\displaystyle{\lim_{x \to a^+} f(x)}$, and then we compare the three quantities.

Example: If $f(x) = \begin{cases} 1-x & x< 0, \cr x^2 & x \ge 0, \end{cases} \qquad $ then 

Then $\displaystyle\lim_{x \to 0^-} f(x)= \lim_{x \to 0^-} (1-x) = 1,\qquad\displaystyle\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} (x^2) = 0\qquad$, and  $f(0) = 0^2  = 0. $

DO:  Check that the values above are correct, using the given piecewise definition of $f$.

Since the limits from the left and right do not agree, the limit does not exist, and the function is discontinuous at $x=0$.