In Fourier analysis, while talking about pointwise convergence, we generally start with the class of functions called, BV functions (functions of bounded variation), which have a finite total variation.
I'd like to avoid this type of classification as they are more complex and I don't really need such a huge class of functions for my problems. But I'd like my class of functions to hold all results of BV functions (pertaining to Fourier analysis) in most generic way possible.
So I'd like to define my class of functions $\mathcal{J}(0,1)$ defined on the interval $(0,1)$ as, for any $f\in \mathcal{J}(0,1)$, $f:(0,1)\to\mathbb{R}$ such that $f$ admits only jump discontinuities, and wherever $f$ is continuous, it is also differentiable.
This class of functions inherits all the convergence properties of Fourier analysis in decently generic way and yet excludes the bad guys like the cantor function (Devil's stair case).
My question is, is there any compact name existing in literature for this class of functions?