It’s Professor Dave, let’s learn about continuity in functions.
We’ve just learned about a variety of functions, and up until now, they have all been continuous
functions.
These are functions where there are no gaps whatsoever.
Any X value can act as an input, and we get all the corresponding values for the function
as a continuous curve.
In other words, the function can be drawn without lifting the pencil from the paper.
This is true of lines and parabolas, and some other kinds of functions.
But some functions are discontinuous.
Some of these do not have domains that include all real numbers, there are some X values
that are not valid inputs, and thus they are not represented on the function.
Consider the function one over X minus one.
As we already know, X can’t be equal to one, because then the denominator would be
zero, and the function would be undefined.
That’s why this function, when we graph it, is undefined at X equals one, meaning
the function doesn’t touch the line X equals one.
Instead, X equals one acts as an asymptote.
This is a line that is not part of the function, but instead, the function gets closer and
closer to the line without ever touching it.
We can see that as the function approaches X equals one from the left, smaller and smaller
changes in X are producing greater and greater changes in the function.
This continues towards the limit of negative infinity as X approaches one.
When approaching from the right, it’s essentially the same thing.
The function goes to positive infinity as X approaches one.
We will talk more about asymptotes later, but even when there are no asymptotes, a function
can be discontinuous if there is a jump, or even just a hole in the function.
This could mean a function that appears continuous except that a single point is missing, or
one in which it appears that a portion of it has been shifted in some direction.
Take X squared minus one over X minus one.
If we are clever, we can see that the numerator will factor into binomials, and that these
two terms cancel, leaving us with X plus one.
Therefore, the graph will resemble the line X plus one.
However, if we return to the original function, there is one tiny difference.
Because the denominator can’t equal zero, X can’t equal one, and so one is omitted
from the domain of this function.
That means the line will have one discontinuity, there will be a hole at X equals one where
the function can’t be evaluated.
Lastly, we can examine piecewise functions.
These are when a function is comprised of several pieces, and we evaluate the function
differently depending on the number we are plugging in.
For example, we can have a function that is equal to five when X is less than negative
two, and equal to X minus one when X is greater than or equal to negative two.
That would mean that all the way from negative infinity, the function will be equal to five.
So that’s this horizontal line leading all the way up to the point negative two, five,
with an open circle, meaning that this point is not included.
Then, from negative two until positive infinity, the function will look like the graph of X
minus one.
So we can draw that starting at negative two, negative three, filling that in with a closed
circle, since that point is included in the function.
This extends to positive infinity, so here’s what we end up with.
And that’s what we need to know about continuity in functions, so let’s check comprehension.