Subsection 1.6.1 Continuity
We have seen that computing the limits some functions — polynomials and rational functions — is very easy because
\begin{align*}
\lim_{x \to a} f(x) &= f(a).
\end{align*}
That is, the the limit as \(x\) approaches \(a\) is just \(f(a)\text{.}\) Roughly speaking, the reason we can compute the limit this way is that these functions do not have any abrupt jumps near \(a\text{.}\)
Many other functions have this property, \(\sin(x)\) for example. A function with this property is called “continuous” and there is a precise mathematical definition for it. If you do not recall interval notation, then now is a good time to take a quick look back at Definition 0.3.5.
Definition 1.6.1
A function \(f(x)\) is continuous at \(a\) if
\begin{align*}
\lim_{x \to a} f(x) &= f(a).
\end{align*}
If a function is not continuous at \(a\) then it is said to be discontinuous at \(a\text{.}\)
When we write that \(f\) is continuous without specifying a point, then typically this means that \(f\) is continuous at \(a\) for all \(a \in \mathbb{R}\text{.}\)
When we write that \(f(x)\) is continuous on the open interval \((a,b)\) then the function is continuous at every point \(c\) satisfying \(a \lt c \lt b\text{.}\)
So if a function is continuous at \(x=a\) we immediately know that
- \(f(a)\) exists
- \(\ds \lim_{x \to a^-}\) exists and is equal to \(f(a)\text{,}\) and
- \(\ds \lim_{x \to a^+}\) exists and is equal to \(f(a)\text{.}\)