Continuity

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Anything You Want to Talk About?

Especially related to one-sided limits, infinite limits, or other things from classes you couldn’t come to in person?

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Misc

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The Math Learning Center is open virtually. See yesterday’s announcement for details.

(Dis)Continuity

From “Continuity at a Point” and “Types of Discontinuities” in section 2.4 of the textbook, and this discussion.

Part 1

Here is a sketch of the graph of a hypothetical function. Where is this function discontinuous? What requirement(s) for continuity are missing at each discontinuity? What kind of discontinuity is each? Give one or two examples of places where this function is continuous. (Interpret each tick mark on the x axis as corresponding to 1 unit if it helps you identify points on the graph.)

Answers in the discussion are generally good at identifying where there are discontinuities, and at identifying the types of discontinuity. There was less attention to relating the discontinuities to specific elements in the definition of “continuous” that failed to hold (while the discontinuities tend to be visually obvious, working with definitions is an important part of mathematical thinking, because everything in math has a reason, generally traceable back to some definition.) Here’s the above graph (with an extension to illustrate one last kind of discontinuity), highlighting the missing elements of the definition:

For the discontinuity at x = -2 (the leftmost one in the graph), the limit exists, but the function doesn’t have a value.

For the discontinuity at x = -1, the function has value but the limit doesn’t exist. Each 1-sided limit exists, but they aren’t equal.

(There’s a discontinuity at x = 2, where the function has a vertical asymptote. We didn’t mark this in the picture, but noted that neither the function nor its limit are defined here, because even though we write such things as “lim_{x → 2} f(x) = ∞,” that’s just a shorthand for saying the limit doesn’t exist for a particular reason. Infinity is not a number!)

For the discontinuity at x = 4, neither f(x) nor its limit exist.

For the discontinuity we added at x = 6, the limit does exist, and f(x) is defined, but they don’t equal each other.

Part 2

Consider the function

What does the graph of this function look like?

This graph is interesting because it has not just a point hole where the function is undefined, but a huge gap between x = -1 and x = 1. This also brings up the observation that we (meaning everyone who teaches calculus, writes calculus books, etc.) somewhat arbitrarily decide that we will talk about the calculus of functions whose values and arguments are real numbers, even though we could apply calculus ideas to complex numbers and functions — take “Complex Analysis” some day if you want to pursue this idea.

Is this function continuous or discontinuous at t = 0? Discontinuous, because x = 0 is in the middle of a gap where the function doesn’t have real values.

How about at t = 1? The function is defined here, since g(1) = 0, but the limit isn’t. This is another example of a place where the limit from one side — the right in this case — is defined, but the limit from the other doesn’t exist. (There is a question and discussion of this possibility in yesterday’s class notes also.)

Give some examples of x values where this function is continuous, and show why it is continuous at those values. Everywhere except the interval [-1,1], because at all other points the function and its limit are both defined, and equal to each other.

Next

Continuity over intervals and the Intermediate Value Theorem.

Read “Continuity over an Interval” and “The Intermediate Value Theorem” in section 2.4 of the textbook.

Participate in this discussion by class time Friday.

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