A continuous function has no breaks in the domain of its independent variable.
This definition implies the entire domain of the independent variable. We can restrict the domain to some interval
and call a function continuous in that interval if it has
no breaks in that interval. If we don not specify this interval then we mean the entire domain.
Consider the graph below. In the interval about x = c there are no
breaks in f(x). So, in this interval f(x) is continuous.
In the interval about x = d there is a break in f(x), that
is, there is a hole. In this interval f(x) is not continuous, that is, it is discontinuous
Since there is a break in f(x), f(x) is not continuous.
The function in the next graph is continuous. A function does not need to be "smooth," it can have jagged edges such as this one.
Now we see a function that jumps at x = c. This is a break in the function
and therefore this function is not continuous in this interval.
A function f(x) is continuous at x = c if and only if
Note the words "if and only if." These words mean that if the function is continuous
at x = c then the equation is valid. Now the other way, if the equation is valid then the
function is continuous. We shorten "if and only of" to "iff."
For example, in the last graph the
limit at x = c does not exist because the one-sided limits are unequal. Therefore the equation is
invalid, and the function is therefore discontinuous.
In the first two examples at x = c, the one-sided limits are the same and evaluate to
f(c), therefore those functions are continuous.
So, if at some point c, the function's value is not f(c) then it cannot be continuous at c.
Also, if the limit exists of the function at x = c but does not equal f(c) then it cannot be continuous at
x = c.
This is the case for all of the graphs we've seen that contain breaks and holes.
If we limit the range to one side of a point we can still talk about continuity. Recall that
a limit in general that exists requires the left side and right side limits to exist and be equal.
Well, if we focus only on the left side then we can talk about a function being continuous from the left,
and if we focus only on the right side then we can talk about a function being continuous from the right.
Mathematically we would write:
f(x) is continuous from the left iff:
f(x) is continuous from the right iff:
Consider the following graph:
This function is continuous from the left as x approaches 4 since the limit approaches 7 and equals the value f(4) = 7.
The function is discontinuous from the right as x approaches 4 since the limit approaches 4 but f(4) = 7.
This function is continuous from the left as x approaches 9 since the limit approaches 6 and equals the f(9) = 6.
The function is discontinuous from the right as x approaches 9 since the limit approaches 7 but f(9) = 6.
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