Kink versus jump

Question

Suppose I have a function defined as follows $n(r) = \begin{cases} n_h \quad \text{if} \quad r< \hat{r} \\ n_l \quad \text{if} \quad r \ge \hat{r}\end{cases}$. Where $n_h > n_l$ and $r \in [0,\infty)$. I want to understand how the product $n(r) \cdot r$ function will look like? On one hand I think there will be a jump at the cut off $\hat{r}$ in the product, simply because I am multiplying a continuous function with a 'jump' function. On the other hand I feel since only the slope is changing I should just have a "kink" at the point $\hat{r}$. To me the first reason sounds more convincing but then I cannot convince myself of a case when there will just be a kink and no jump in the product (i.e. what kind of $n(r)$ function will produce a kink and not a jump in the $n(r) \cdot r$ function?). Sorry for what might be a silly question. I feel at times I should have been much better at real analysis. Any help will be much appreciated.

Answer 1

There are two cases in which there is only a kink: that where $n_h=n_l$, and that where $\hat r=0$, as otherwise the limit of $n(x)$ as $x$ approaches $\hat r$ from below will be $n_h\hat r\neq n(\hat r)=n_l\hat r$, so there will be a jump.