Real Rational Function is Continuous

Theorem

A real rational function is continuous at every point at which it is defined.

Thus a real rational function is continuous on every interval of $\R$ not containing a root of the denominator of the function.

Let:

$\map R x = \dfrac {\map P x} {\map Q x}$

be a real rational function, defined at all points of $\R$ at which $\map Q x \ne 0$.

Let $c \in \R$.

$\ds \lim_{x \mathop \to c} \map P x = \map P c$

and:

$\ds \lim_{x \mathop \to c} \map Q x = \map Q c$

$\ds \lim_{x \mathop \to c} \map R x = \lim_{x \mathop \to c} \frac {\map P x} {\map Q x} = \frac {\map P c} {\map Q c}$

whenever $\map Q c \ne 0$.

So whatever value we choose for $c$ such that $\map Q c \ne 0$, we have that $\map P x$ is continuous at $c$.

From the definition of continuity on an interval, the second assertion follows.

$\blacksquare$