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The limit of a continuous function at a point is equal to the value of the function at
Limits are simple to compute when they can be found by plugging the value into the
function. That is, when We call this property continuity.
A function is continuous at a point if
Consider the graph of the function
Which of the following are true?
is continuous at is continuous at is
It is very important to note that saying
“a function is continuous at a point ”
is really making three statements:
is defined. That is, is in the domain of .
The first two of these statements are implied by the third statement.
Find the discontinuities (the points where a function is not continuous) for the
function described below:
To start, is not even defined at , hence cannot be continuous at .
Next, from the plot above we see that does not exist because Since does not exist,
cannot be continuous at .
We also see that while . Hence , and so is not continuous at .
Building from the definition of continuity at a point, we can now define what it
means for a function to be continuous on an open interval.
A function is continuous on an open interval if for all in .
Loosely speaking, a function is continuous on an interval if you can draw the
function on that interval without any breaks in the graph. This is often referred to as
being able to draw the graph “without picking up your pencil.”
Continuity of Famous Functions The following functions are continuous on their
natural domains, for a real number and a positive real number:
Sine and cosine
In essence, we are saying that the functions listed above are continuous wherever they
The function is of the form for a positive real number . Therefore, is
continuous for all positive real values of . In particular, is continuous at . Since is
continuous at , we know that . That is, .
The function is a constant. Therefore, is continuous for all real values
of . In particular, is continuous at . Since is continuous at , we know that . That is, .
The function is contionuous for all real values of . In particular, is
continuous at . Since is continuous at , we know that . That is, .
Left and right continuity
At this point we have a small problem. For functions such as , the natural domain is .
This is not an open interval. What does it mean to say that is continuous at
when is not defined for ? To get us out of this quagmire, we need a new
A function is left continuous at a point if .
A function is right continuous at a point if .
This allows us to talk about continuity on closed and half-closed intervals.
A function is
continuous on a closed interval if is continuous on , right continuous
at , and left continuous at ;
continuous on a half-closed interval if is continuous on and right
continuous at ;
continuous on a half-closed interval if is is continuous on and left
continuous at .
Here we give the graph of a function defined on .
Select all intervals for which the following statement is true.
The function is continuous on the interval .
Notice that our function is left continuous at so we can include in the
interval . Four is not included in the interval because our function is not right
continuous at . Similarly, our function is neither right or left continuous at
, so is not included in any intervals. Our function is left continuous at
and right continuous at so we included these endpoints in our intervals.