Why Huber loss has its form?

Question

Huber loss formula is

$\hspace{3.0cm} L_\delta(a) = \begin{cases} \frac{1}{2} a^2 && |a| \leq \delta \\ \delta (|a| - \frac{1}{2} \delta) && |a| > \delta\end{cases}$ where $a = y - f(x)$

As I read on Wikipedia, the motivation of Huber loss is to reduce the effects of outliers by exploiting the median-unbiased property of absolute loss function $L(a) = |a|$ while keeping the mean-unbiased property of squared loss function $L(a) = a^2$ within the typical region (i.e. region without outliers)

I don't know why don't Huber loss has the form:

$\hspace{3.0cm} L_\delta(a) = \begin{cases} \frac{1}{2} a^2 && |a| \leq \delta \\ c |a| && |a| > \delta\end{cases}$ where $c$ is some constant which controls the error magnitude of outliers

or similar forms ?