On 11/11/2012 7:16 AM, Joe Niederberger wrote: > Excuse me, but I feel the need to clarify the post copied below. Clyde was saying his description of "zooming" and "convergence" (see below) was the "common sense view" of a continuous functions, to which I say WHAT? > > The common sense view of a continuous function is as I sate further below -- the graph of which I can draw without lifting the pencil. I think the conversion of that intuitive view into a definition involving limits was a great achievement for mathematics, and took some time to hit upon, nothing common sense about it. One must first struggle with Zeno's paradox to appreciate it. > > Joe N But we should always check that our common sense is indeed sensible. Consider the function:
F: y= x^2 if x is rational; -x^2 if x is irrational
Appears to be continuous at a single point (0,0) only. Can be drawn without lifting the pencil - as long as the pencil stays still.