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:
1.
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 ______________.
2.
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 ___________________.
3.
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 __. The formal definition of continuity requires that the function be
defined at the x value in question.
4.
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 _______________ at at x = 1.
5.
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 __________________ at x = 1.
6.
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 it is
____________________ at x = 1.