Here are the details for those who might feel too lazy to chase the links in the comments above.

The *Hinge loss* for $x \in R$ is defined as
$$H(x) = \max(0, 1-x)$$

Jason Rennie in his article "Smooth Hinge Classification" describes the following *smooth* version of the Hinge loss (a smoothed version was being sought because of discontinuity in the derivative at $x=1$).

Rennie defines (the definition seems natural enough that somebody might have also found a similar one; I will be happy to be corrected) the *smoothed Hinge loss*:
$$H_s(x) = \begin{cases} \tfrac{1}{2}-x & x \le 0,\\\\
\tfrac{1}{2}(1-x)^2 & 0 < x < 1\\\\
0 & x \ge 1
\end{cases}$$

This loss is smooth, and its derivative is continuous (verified trivially).

Rennie goes on to discuss a parametrized family of smooth Hinge-losses $H_s(x; \alpha)$. Additionally, several other variations are possible, depending on what numerical behavior seems more appropriate for an application.