Neither Left Nor Right Continuous Continuous Only From the Left

Continuity is defined by limits.

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 below

Which of the following 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, but are important enough that we want to break them up to highlight their significance.

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 the given intervals for a real number and a positive real number:

Constant function

is continuous on .

Identity function

is continuous on .

Power function

is continuous on .

Exponential function

is continuous on .

Logarithmic function

is continuous on .

Sine and cosine

Both and are continuous on .

In essence, we are saying that the functions listed above are continuous wherever they are defined, that is, on their natural domains.

Compute:

The function is of the form for a real number . Therefore, is continuous 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 definition:

A function is left continuous at a point if .

A function is right continuous at a point if .

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.

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 .

What are the largest intervals of continuity for this function?

and , , and , , and , , and , , and , , and , , and

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.

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Source: https://ximera.osu.edu/calcwithreview/readingAssignments/readingLL/limitLaws/digInContinuity

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