Continuity: A 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.
This device cannot display Java animations. The above is a substitute static image
1. A continuous function
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.
2. A sine function
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.
3. Essential discontinuity
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.
4. A jump discontinuity
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 lefthand and righthand limits are different). Hence
this function is not continuous at at x = 1.
5. Removeable discontinuity
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.
6. Displaced point
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.
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Other differentiation topics
Acknowledgements
Derived from the work of Thomas S. Downey under a Creative Commons Attribution 3.0 License.
