(Did you receive my first e-mail about this?) P.S. I was wrong (in that first e-mail) and Andy's right: Your function is continuous at each irrational number. Sorry about that. --Dan On 2013-08-10, at 8:10 AM, Marc LeBrun wrote:
Dumb questions I’m not sure even how to ask: could someone help describe or visualize the following function f?
For rational x = a/b (in lowest terms) let f(x) = a/(b+1)
So for instance we can say things like x/2 <= f(x) < x.
Next, to extend f to real x, we observe that in sequences of rationals x_n “converging on” x the denominators (almost always) grow, so that f(x_n) “approaches” x. So let’s define f as the identity on irrational values.
What else can we say about f? Is it continuous? What about its derivative? Fourier transform? It’s sure “bumpy”, but is it “fractally”?
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