Comment Comment by seanhunter The Weierstrass function is cool but the undisputed champion of calculus counterexamples has to be the Dirichlet function[1]f(x) = 1 if x is rational, 0 otherwise.It is defined over all real numbers but continuous nowhere. Also if you take the Dirichlet function and multiply it by x so you getg(x) = x if x is rational, 0 otherwise…then you have something that is continuous at exactly one place (0) and nowhere else, which also is pretty spectacular.[1] https://mathworld.wolfram.com/DirichletFunction.html parent seanhunter 1 year ago
f(x) = 1 if x is rational, 0 otherwise.
It is defined over all real numbers but continuous nowhere. Also if you take the Dirichlet function and multiply it by x so you get
g(x) = x if x is rational, 0 otherwise
…then you have something that is continuous at exactly one place (0) and nowhere else, which also is pretty spectacular.
[1] https://mathworld.wolfram.com/DirichletFunction.html