2.4: Continuity
( \newcommand{\kernel}{\mathrm{null}\,}\)
Continuity
A function that is "friendly" and doesn’t have any breaks or jumps in it is called continuous. More formally,
A function f is continuous at x=a if and only if limx→af(x)=f(a).
The graph below illustrates some of the different ways a function can behave at and near a point, and the table contains some numerical information about the function and its behavior.

a | f(a) | limx→af(x) |
1 | 2 | 2 |
2 | 1 | 2 |
3 | 2 | Does not exist (DNE) |
4 | Undefined | 2 |
Based on the information in the table, we can conclude that f is continuous at 1 since limx→1f(x)=2=f(1).
We can also conclude from the information in the table that f is not continuous at 2 or 3 or 4, because limx→2f(x)≠f(2), limx→3f(x)≠f(3), and limx→4f(x)≠f(4).
The behaviors at x=2 and x=4 exhibit a hole in the graph, sometimes called a removable discontinuity, since the graph could be made continuous by changing the value of a single point. The behavior at x=3 is called a jump discontinuity, since the graph jumps between two values.
So which functions are continuous? It turns out pretty much every function you’ve studied is continuous where it is defined: polynomial, radical, rational, exponential, and logarithmic functions are all continuous where they are defined. Moreover, any combination of continuous functions is also continuous.
This is helpful, because the definition of continuity says that for a continuous function, limx→af(x)=f(a). That means for a continuous function, we can find the limit by direct substitution (evaluating the function) if the function is continuous at a.
Evaluate using continuity, if possible:
- limx→2x3−4x
- limx→2x−4x+3
- limx→2x−4x−2
Solution
- The given function is polynomial, and is defined for all values of x, so we can find the limit by direct substitution:limx→2x3−4x=23−4(2)=0.
- The given function is rational. It is not defined at x=−3, but we are taking the limit as x approaches 2, and the function is defined at that point, so we can use direct substitution:limx→2x−4x+3=2−42+3=−25.
- This function is not defined at x=2, and so is not continuous at x=2. We cannot use direct substitution.