curvature is continuous function

Question

Let $\gamma :(a,b)\to \mathbb{R}^n$ be a regular curve. Let $\kappa(t)$ denotes the curvature of the curve at point $t$.Then

1. $\kappa$ is a continuous function.
2. If for every $t$, $\kappa(t)>0$ then it is a smooth function.

Since, $\kappa(s)=\|\tilde{\gamma}''(s)\|, $ where $\tilde{\gamma}$ is a unit speed parameterization. So $(1)$ is pretty much clear as it is the composition of two continuous function, namely $\mathbf{x}\mapsto \|\mathbf{x}\|$ and $\tilde{\gamma}''$.

But for the second one I am not sure what to do. Also I am not sure why, In general, curvature is not a smooth function.