Precision and Loopholes in Definitions and Theorems: 2 Questions

By far the best benefit of having honors or AP classes is the questions students ask and the ones I become driven to ask.

We use the Larson Hostetler Edwards textbook for Calculus.

Question 1:

Here is their definition of strictly monotonic and a theorem that uses the concept.



A student asks, what about the graph of this function? It isn't one-to-one, but isn't this strictly monotonic based on that definition?



This problem seems to ask the same question.



My burning question right now, is continuity a condition for the definition of strictly monotonic? If so, why doesn't the textbook mention it? Or did I miss it somewhere perhaps?

I told my students I would look into it more, and I can't seem to find anything that directly answers the question.

Question 2:

This question appears on a chapter exam I used that had been written by a previous teacher.



When keying the exam, I did it without looking at a graph, and said b) because when you substitute, you get arctan (1/0). Most of the students said a), so I looked at the graph, and I see why.



I was trying to find some more precise definition of a jump discontinuity, but I am having trouble reconciling this one. I notice that the limits approach +/- pi/2, since the range of arctan x is (-pi/2, pi/2).

Is this a badly worded question, or am I missing something with how the definitions and theorems apply? Thanks in advance for the help!