Infinite length

[Tolaso J Kos]

Show that the curve $\displaystyle \alpha(t)= \left\{\begin{matrix}
\left ( t, t \sin \left ( \frac{\pi}{t} \right ) \right ) & , & t\neq 0 \\
0&, &t = 0
\end{matrix}\right.$ has infinite length on $[0, 1]$.

[Demetres]

We see that at $t = 1/(2n)$ the curve is at $(1/(2n),0)$ while at $t=2/(4n+1)$ the curve is at $(2/(4n+1),2/(4n+1)$ so the total length of the curve in the interval $[2/(4n+1),2/(4n)]$, by the triangle inequality, is at least $2/(4n+1)$. Summing up over all positive integers $n$ we get from the divergence of the harmonic series that the length of the curve must be infinite.