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PMA Rudin p.141

Rudine defined a curve $\gamma_3:[0,2\pi]\rightarrow \mathbb{C}:t\mapsto e^{2\pi i t sin(\frac{1}{t})}$.

It reallt doesn't make sense to define such a function since it is not defined at $t=0$.

What's actually Rudin intended?

This is how i guessed.

Let $\alpha$ be a continuous fuction defined on $(a,b]$.

Then define length of $\alpha$ as $\lim_{s\to a} \Lambda(\alpha\upharpoonright [s,b])$.

Then this problem makes sense.

However, is this definition generally used? Is there another widely being used definition of 'length of a curve on an open interval'?

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Since $\gamma_3(t)=\exp(2\pi \mathrm i t \sin(1/t))$ for every $t\ne0$ and $t\sin(1/t)\to0$ when $t\to0$, it seems pretty clear that Rudin intended $\gamma_3(0)=1$.