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.


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:

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

and:

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

Thus by Quotient Rule for Limits of Real Functions:

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

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.

$\blacksquare$


Sources

  • 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: $\S 1.4$: Continuity
  • 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 8.13$
  • 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): continuous function (v)
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