#math/calculus A function $f$ is **continuous at $x_{0}$** when $ \lim_{x\to x_{0}}f(x)=f(x_{0}). $ Equivalently, $f(x_{0})$ is defined, the limit exists, and the two values agree. A function is continuous on an interval when it is continuous at every point of that interval. A **rational function** has the form $r(x)=p(x)/q(x)$, where $p$ and $q$ are [[2A2 - Polynomials|polynomials]] and $q$ is not the zero polynomial. Polynomials are continuous everywhere, and rational functions are continuous wherever their denominators are nonzero. For example, $ r(x)=\frac{x^{2}+x-6}{x^{2}+2x-8} =\frac{(x+3)(x-2)}{(x+4)(x-2)} =\frac{x+3}{x+4}\quad(x\neq 2). $ Although $r(2)$ is undefined, the replacement rule gives $\lim_{x\to 2}r(x)=5/6$. This is a **removable discontinuity**: defining $r(2)=5/6$ would make the function continuous there. > [!Practice] > Marsden & Weinstein, p. 69, starred exercises.