Robert Hansen says: >But "a function whose graph..."? I think someone would have to understand a function first, right?
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.
Another starting point would be simply to say "no gaps or jumps". Yet another would be more like Zeno - to get from A to B I must traverse and infinite set of intermediate points. In particular, wherever I am I can only make a tiny amount of progress given a tiny amount of time.
At the end of the day, one will have modified and filled in one's understanding, but to suppose one must necessarily start from the Weierstrass definition is absurd in my opinion.
Robert Hansen says: Formal Reasoning: What does that actually mean and why is it significant?
On the other hand one can easily find examples of what is commonly termed *informal* reasoning and informal logic. I have never read any description of a mathematician or scientist working exclusively in formal mode. Informal methods would seem to be the rule for development of ideas and making advances, formal for testing, writing up results and presentation.