Real Rational Function is Continuous

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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.


Proof

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 $c \in \R$.

From Real Polynomial Function is Continuous‎:

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

and:

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

Thus by Quotient Rule for Limits of Functions:

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

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.

$\blacksquare$


Sources