# Question about the continuity of the length of a continous parametrized curve

Any hint or demo to prove that the length of a continous parametrized curve defines a continous function in a normed space?

Define $$f$$ by $$x\mapsto x^2\sin(1/x)$$ if $$x\neq0$$ and $$x\mapsto0$$ if $$x=0$$. Note that $$f$$ is differentiable. However, its length $$\int_\Omega\sqrt{1+\left(f'\left(x\right)\right)^2}\text{d}x$$ is undefined on a closed interval $$\Omega$$ containing the origin and therefore not continuous.
• @JackTalion Fine. Think of it as $\gamma(t)=(t,f(t))$. – wjm Mar 17 at 22:08