Smooth parametrisation in the complex plane. My book defines a complex smooth parametrisation like this. First a parametrisation is a complex funtion z of a real variable t. Where t is defined on $[a,b]$. It is smooth if $z'(t)$ exists and is continuous. They also say that $z'(t) \ne 0$ for $t \in [a,b]$. I am wondering why we have this last condition? Does it have anything to do with how we define the integral later. Or can it graphically be shown that the function is not smooth if it's derivative is 0 at a point? Also I see that if the derivstive is 0 at an interval then the parametrisation may stand still when t vary, but if it happens only in a point the function does not stand still? But no matter if it is in a point or an interval it is zero the function still looks smooth?

Actually there are many reason to ask $z'(t)\
eq 0$.

It there is such a point on your complex function, then many things can happen, in particular the function can break at some point and being unpredictable. It means that at this very bad point, you can't guess where the curve is going next: there is no linear approximation of the curve, no tangent line or direction at this point. You also can't speak about the curvature or even find a unit speed parametrization of the curve.

In fact, begin smooth means that, locally, the curve should looks like a straight line. This is a very import notion of submanifold of $\mathbb R^n$.

Here is an example of a curve $t\mapsto z(t)=(x(t),y(t))$ with one bad point at $(0,0)$ with $$x(t)=\dfrac{t^2}{1+t-t^3}, \quad y(t)=\ln(1+t^2)$$ !enter image description here