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When I first saw a "proof" like this, the explanation of its incorrectness was something like "limits don't preserve curve length". I wasn't satisfied with that
by jbapple 16y ago
When I first saw a "proof" like this, the explanation of its incorrectness was something like "limits don't preserve curve length". I wasn't satisfied with that answer until I took a first course in real analysis, which explained (some of) the reasons behind calculus.
Real analysis was very satisfying, in somewhat the same way that building low-level software or libraries is satisfying -- I got to understand the guts. It was also fun to learn about erroneous historical assumptions made due to insufficient rigor. IIRC, until at least the 1870s, it was believed that any continuous function must be differentiable almost everywhere, and that in fact this should be obvious. It turns out that one can construct continuous functions that are nowhere differentiable!
This was one of my favorite topics in the class:
http://en.wikipedia.org/wiki/Construction_of_the_real_numbers http://en.wikipedia.org/wiki/Construction_of_the_real_number...
- xyzzyz 16y agoWhat is even more tricky is that the set of functions differentiable in at least one point is first category, and thus nowhere differentiable continuous functions are dense in continuous functions space.
- drbaskin 16y agoNot only are they dense, but they are "most" continuous functions! (For an appropriate interpretation of "most", of course)
- lmkg 16y ago> Real analysis was very satisfying, in somewhat the same way that building low-level software or libraries is satisfying -- I got to understand the guts. That is exactly the reason why taking a course in real analysis made me switch majors. I was trying to get that feeling from physics, but only found it to my satisfaction in math.