# 2.4 | Continuity and One-Sided Limits

Let $$f$$ be a function defined on an open interval containing all values of $$x$$ close to $$a$$. Then $$f$$ is continuous at $$a$$ if
$$\lim_{x \rightarrow a} f(x) = f(a)$$
If $$f$$ is defined for all values of $$x$$ close to $$a$$ but the equation above is not satisfied, then $$f$$ is discontinuous at $$a$$ or $$f$$ has a discontinuity at $$a$$.
Let $$f$$ is defined for all values of $$x$$ close to $$a$$ with
$$\lim_{x \rightarrow a^-} f(x) = L$$
and
$$\lim_{x \rightarrow a^-} f(x) = M.$$
If $$L$$ and $$M$$ are finite and $$L \not = M$$ then $$f$$ has a jump discontinuity at $$a$$.
Let $$f$$ is defined for all values of $$x$$ close to $$a$$ with
$$\lim_{x \rightarrow a} f(x) = L.$$
If $$L$$ is finite and $$L \not = f(a)$$ then $$f$$ has a removable discontinuity at $$a$$.
Let $$f$$ is defined for all values of $$x$$ close to $$a$$. If either
$$\lim_{x \rightarrow a^-} f(x) = \pm \infty$$
or
$$\lim_{x \rightarrow a^+} f(x) = \pm \infty$$
then $$f$$ has an essential discontinuity at $$a$$.
A function $$f$$ is continuous from the right at $$a$$ if
$$\lim_{x \rightarrow a^+} f(x) = f(a)$$
A function $$f$$ is continuous from the left at $$a$$ if
$$\lim_{x \rightarrow a^-} f(x) = f(a)$$
A function $$f$$ is continuous on an open interval $$(a,b)$$ if it is continuous at every number in the interval. A function $$f$$ is continuous on a closed interval $$[a,b]$$ if it is continuous on $$(a,b)$$ and is also continuous from the right of $$a$$ and from the left of $$b$$. A function $$f$$ is continuous on a half-open interval $$[a,b)$$ or $$(a,b]$$ if it is continuous on $$(a,b)$$ and $$f$$ is continuous from the right of $$a$$ or $$f$$ is continuous from the left of $$b$$, respectively.
If the functions $$f$$ and $$g$$ are continuous at $$a$$, then the following functions are also continuous at $$a$$.

• $$f \pm g$$
• $$fg$$
• $$cf$$, where $$c$$ is any constant
• $$\frac{f}{g}$$, if $$g(a) \not = 0$$
• A polynomial function is continuous on $$(-\infty, \infty)$$.
• A rational function is continuous on its domain.
The functions $$\sin x, \cos x, \tan x, \sec x, \csc x,$$ and $$\cot x$$ are continuous at every number in their respective domains.
If $$f$$ is continuous on its domain, then $$f^{-1}$$ is continuous on its domain. Also, the functions $$\sin^{-1} x, \cos^{-1} x, \tan^{-1} x, \sec^{-1} x, \csc^{-1} x, \cot^{-1} x, a^x,$$ and $$\log_a x$$ are continuous on their respective domains.
If the function $$f$$ is continuous at $$L$$, then
$$\lim_{x \rightarrow a} f(g(x)) = f\left(\lim_{x \rightarrow a}g(x) \right)$$
If the function $$g$$ is continuous at $$a$$ and the function $$f$$ is continuous at $$f(a)$$, then the composition $$f \circ g$$ is continuous at $$a$$.
The Intermediate Value Theorem: If $$f$$ is continuous on a closed interval $$[a,b]$$ and $$M$$ is any number between $$f(a)$$ and $$f(b)$$, inclusive, then there is at least one number $$c$$ in $$[a,b]$$ such that $$f(c) = M$$.
If $$f$$ is a continuous function on a closed interval $$[a,b]$$ and $$f(a)$$ and $$f(b)$$ have opposite signs, then the equation $$f(x) = 0$$ has at least one solution in the interval $$(a,b)$$ or, equivalently, the function $$f$$ has at least one zero in the interval $$(a,b)$$.