Let $f(t)=t \sin \frac 1 t$ if $0 < t \le 1$, and $f(t)=0$ if $t = 0$.
Consider the curve $\gamma$ with parametric representation $\phi : [0, 1] → \mathbb{R}^2$ defined by $\phi(t) = (t,f (t))$. Prove that $\gamma$ is a continuous curve which is not rectifiable.
for continuity, do I just show that $t\sin\frac{1}{t}$ is continuous on the interval given? To prove it is not rectifiable, I have to show that length of $\phi$ is infinite. Do I show the integral of the derivative of $t \sin \frac 1 t$ is infinite?