Definition of smooth curve

Question

Definition of smooth curve: A parametric curve $\mathrm Z(t)=x(t)+i > y(t)$ on $[a,b]$ is called smooth if

$\mathrm Z'(t)=x'(t)+i y'(t)$ exists and continuous on $[a,b]$.

$\mathrm Z'(t)$ is non zero on $(a,b)$.

But my problem is that why is the word "continuous" written? As We know that differentiability implies continuity.

score 5 · Answer 1 · answered Jan 31 '19 at 02:09

5

Differentiability of $Z(t)$ implies continuity of $Z(t)$, but does not imply the derivative $Z'(t)$ is continuous, which is what the definition says.

answered Jan 31 '19 at 02:09

obscurans

3,422

How is this? Please provide me a proof of this statement. – Mathforjob Jan 31 '19 at 02:28
2

https://math.stackexchange.com/questions/1391544/differentiable-but-not-continuously-differentiable – obscurans Jan 31 '19 at 02:30
My doubt remains as it is. Provide me a simple counter example. – Mathforjob Jan 31 '19 at 02:44
2

Most "simple" (to define) functions are analytic. One single counterexample proves the statement "does not imply". You are free to search for more examples on your own. – obscurans Jan 31 '19 at 02:49
Thank you so much – Mathforjob Jan 31 '19 at 03:11

Definition of smooth curve

1 Answers1