Continuity: Formal Approach
Now that we have a formal definition of limits, we can use this to define
continuity more formally. We can define continuity at a point on a function
as follows:
The function f is continuous at x = c if f (c) is defined and if 
.
In other words, a function is continuous at a point if the function's
value at that point is the same as the limit at that point. We can use this
definition of continuity at a point to define continuity on an interval as
being continuous at every point in the interval.
Try the following:
- The first graph shown, a simple parabola. Move the slider to pick an x value. Notice that the value of the function, given by y =, is the same as the limit at that point. So the function is continuous
at that x value. Since this is true for any x value that
you pick, the function is continuous everywhere.
- Select the second example from the drop down menu. The sine curve has
more wiggles in it, but it is still continuous. Move the slider to pick
an x value. Like the previous example, everywhere you look the
output value of the function is the same as the limit, so this function
is also continuous everywhere.
- Select the third example. This function has a vertical asymptote at x = 1. Is the function continuous at x = 1? Since the
function isn't even defined there, the answer is no. The formal
definition of continuity requires that the function be defined at the x value in question.
- Select the fourth example. This function jumps from 1 to 2 at x = 1. Notice that f (1) = 2, but the limit at x = 1 does not
exist (because the left-hand and right-hand limits are different). Hence
this function is not continuous at at x = 1.
- Select the fifth example. This function has a hole in it at x = 1. This
time, the limit is defined at x = 1 (and is 1), but the function
does not have a value there, so it is not continuous at x = 1.
- Select the sixth example. This function has a displaced point at x = 1.
This time, the limit is defined at x = 1 (and is 1), the function
does have a value there (f (1) = 2), but the limit and the
function's value are different, so again it is not continuous at x = 1.
This work by Thomas S. Downey is licensed under a Creative Commons Attribution 3.0 License.