Continuous functions are functions that do not have any breaks or jumps in the graph. Mathematically, several essential properties exist with continuous functions, but this section is only interested in identifying if a function is continuous or not.
Three criteria need to be satisfied for a function to be continuous. They are:
The limit of the function has to exist. So,
The value of the function has to exist. So,
And the limit of the function and the value of the function has to be the same value, or:
Take the following function:
Looking at the graph of the function which is below:
The figure shows that this function has two points of discontinuity, and they are at x = -1, and x = 2.
Looking at each of these points individually we can determine why the function is discontinuous at those values.
x = -1
Taking the limit of the function as x approaches -1 finds that the limit does not exist because as the function approaches -1 from the positive direction it approaches ∞, and as the function approaches -1 from the negative direction it approaches -∞. Since ∞ ≠ -∞ the limit does not exist. Since the limit does not exist then the function is not continuous at x = -1.
x = 2
Based on the graph above, f(x) does approach a specific value as x approaches 2. To find what that value is we plug in the value of 2 into the equation.
Because we got the value of 0/0 l’Hopital’s Rule Applies. Taking the derivative of both the numerator and the denominator gives us:
Plugging in the value of 2 into the equation
So, the limit of the function as the value approaches 2 is 2. However, the value of f(2) does not exist, and so the function is not continuous.
x = 0
To determine if the function is continuous at x = 2, first determine the value of f(2). Plugging 2 into the equation yields:
Using the above graph also shows that the limit of the function as x approaches 0 is also 4. So, since both the limit and the value of the function equals 4, then the function is continuous at x = 0.