# Continuity > [!idea]+ Continuity in Laymen's Terms > The graphs of many functions can be traced with a pencil *without lifting it* from the page, this property of a function would make it **continuous**. > > Other functions may have points where a break occurs in the graph while still containing continuous intervals in other sections of the domain. > > Functions that have these breaks can be described as having **discontinuity at a point**. --- > [!abstract]+ Defining Continuity > A function $f(x)$ is **continuous at a point** $a$ if and only if the following three conditions are satisfied: > > i. $f(a)$ is defined > ii. $\lim_\limits{x \to a} f(x)$ exists > iii. $\lim_\limits{x \to a} f(x)=f(a)$ > > A function is **discontinuous at a point** $a$ if it fails to be continuous at $a$. > > > Remember, a point $a$ is **defined** if $f(a)$ has an output value. > > It is **undefined** if no solution is found when using $a$ as an input. A function's continuity can be described in terms of the *entire function*, $f(x)$, or sometimes over a *specific interval* $[a, b]$. - [[Polynomials]] are continuous over their entire domain - [[Notes/Trigonometry]] are continuous over their entire domain. There are multiple [[types of discontinuity]]: 1. Removable (point) 2. Infinite (asymptotes) --- ## Right and Left Sided Continuity As mentioned above, a function's continuity can be described over specific intervals. Here we will take a look at how we'd describe a function's continuity when approaching point the limit from the left and right sides. > [!summary]+ Continuity From the Right/Left > - A function $f(x)$ is said to be **continuous from the right** at point $a$ if $\lim_\limits{x \to a^+} f(x) = f(a)$. > > - A function $f(x)$ is said to be **continuous from the left** at point $a$ if $\lim_\limits{x \to a^-} f(x) = f(a)$. > [!question] Practice: Continuity on an Interval > [PENDING](https://openstax.org/books/calculus-volume-1/pages/2-4-continuity) --- # The Composite Function Theorem The **Composite Function Theorem** allows us to expand our ability to compute limits. Specifically, this theorem ultimately allows us to demonstrate that [[Notes/Trigonometry]] are continuous over their domains. > [!info]+ The Composite Function Theorem > If $f(x)$ is continuous at $L$ and $\lim_\limits{x \to a} g(x) = L$, then > $\lim_{x \to a} f(g(x)) = f(\lim_{x \to a} g(x)) = f(L)$ > [!question] Practice: Continuity of Trigonometric Functions > [PENDING](https://openstax.org/books/calculus-volume-1/pages/2-4-continuity) --- [[The Intermediate Value Theorem]]