Consider the curve $t\to (t,|t|)$ in $\mathbb R^2$. Even if it has a cusp in $(0,0)$ i can reparametrize it with a smooth function. Take for example $$ t\mapsto \begin{cases}(\mathrm e^{-1/t},\mathrm e^{-1/t}) & t>0\\ (-\mathrm e^{1/t},\mathrm e^{1/t}), & t<0\\ (0,0) & t=0 \end{cases} $$

My question is now if i can reparametrize every continuous curve in $\mathbb R^2$ by means of a smooth parametrization? It seems like the preceeding example hints at the possibility to do so for isolated discontinuities. But what about curves like the Weierstrass-function that are nowhere differentiable?

  • $\begingroup$ Each curve which is piecewise $C^r$ ($r>0$) may be reparametrized to a $C^r$-curve. This is certainly not true for only continuos curves. $\endgroup$ – Michael Hoppe Dec 10 '13 at 11:24
  • $\begingroup$ Do you have a reference for this? I mean for the 2nd statement. $\endgroup$ – frog Dec 10 '13 at 11:39
  • $\begingroup$ For example en.wikipedia.org/wiki/Koch_snowflake $\endgroup$ – Michael Hoppe Dec 10 '13 at 11:57

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