Two definitions for a smooth curve equal.

Question

I've encountered these two definitions: 1. $\gamma\colon [a,b]\longrightarrow\mathbb{R^3}$ is smooth if all three derivatives exist and $\gamma^{\prime}(t) \neq 0 \;\; \forall t \in [a,b]$

1. $\gamma\colon [a,b]\longrightarrow\mathbb{R^3}$ is smooth if all three derivatives exist and are continuous.

why they are equivalent?

I mean that why curve has a tangent at each point if the second condition exists?

Thanks in advance.

Answer 1

The definitions are not equivalent. Yes, if you impose the condition that $\gamma'(t)\ne 0$ for all $t$, you'll have a non-zero tangent vector, and hence a tangent line, at each point. For differential geometry, one wants this condition—so that one can reparametrize by arclength, for example.

If you consider $\gamma(t)=(t^2,t^3)$, you see that the curve is a cusp and does not have (classically speaking) a tangent line at $t=0$. Nevertheless, this fits the second definition.