Continuity is defined by limits.

A few sections ago, we discussed what we called ’nice’ functions. These were functions that, at a particular -value , had easy-to-compute limits. In fact, computing the limit of a ’nice’ function is as simple as plugging into the function itself so In other words, if a function is what we called ’nice’ at , the limit of that function at is equal to the function value at . As you might imagine, this property is pretty useful, so we give it a name: continuity.
Consider the graph of below
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Which of the following are true? (Select all that are true)
is continuous at is continuous at is continuous at

It is very important to note that saying

‘‘a function is continuous at a point ’’

is really making three statements:

(a)
is defined. That is, is in the domain of .
(b)
exists.
(c)
.

The first two of these statements are implied by the third statement.

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.

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.’’

Compute:
True or false: If and are continuous functions on an interval , then is continuous on .
True False
True or false: If and are continuous functions on an interval , then is continuous on .
True False
Where is continuous?
for all real numbers at for all real numbers, except impossible to say

Now, we give basic rules for how limits interact with composition of functions. Before continuing on, you may wish to review what is meant by a composition of functions.

Because the limit of a continuous function is the same as the function value, we can now pass limits inside continuous functions.

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 definition:

Now we can say that a function is continuous at a left endpoint of an interval if it is right continuous there, and a function is continuous at the right endpoint of an interval if it is left continuous there. This allows us to talk about continuity on closed intervals.

Here we give the graph of a function defined on .
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What are the largest intervals of continuity for this function?
and , , and , , and , , and , , and , , and , , and