I thought the context here was something like a precalc course - so yeah we can assume "function" is known. The pencil bit is well known and used. Its a starting point for discussion, not a conclusion. If you want to find fault with it I could hardly care less, many people have understood the use of it but some never will.
I am not finding fault with it as a starting point. Like I said earlier, if I were to teach a class in small engine repair, I would start with a small engine.
What I said was that understanding the meaning of drawing a curve without lifting your pencil is NOT the same as understanding the meaning and significance of continuity. The former is a concrete example, the later a theory.
Robert Hansen says:
Formal Reasoning: What does that actually mean and why is it significant?
"Formal Reasoning" is an ill defined term at best.
Not in this conversation. I have been very consistent with the term.
Some people think that physicists and engineers started using complex numbers to represent magnitude and phase in AC circuits because they somehow (common sense I guess) saw that magnitude is real and phase is imaginary. The truth is that 1000's of physicists and electrical engineers had been working with god awful awkward equations for decades until a few (or maybe just one) bright individuals saw how the application of complex numbers simplified all that. Much like Napier saw how logarithms simplified computation.
All of mathematics is developed this way.
There is a layer that I don't think you are taking account of in all this. Take this for whatever. I am really good at solving unique problems, and I can tell you it isn't anything like what you are describing. It flows from the inside out, not the outside in. That is what I want for students.