Quantized space is absolutely discontinuous, and tunneling is a discontinuous system. In fact assuming the universe is quantum it’s discontinuous in reality but the appearance is continuous. But these distinctions aren’t super useful unless you’re dealing with these sorts of effects. Continuity is the approximation, discontinuity is the reality. But depending on what’s useful we use the mathematics that help us.

Tunneling currents are continuous in every parameter, although I admit that when you're dealing with particles you have continuous probability distributions with continuously varying means, rather than continua of matter. (But that should count, because all macroscopic variables are expectation values.)

Specifically at quantized space and time levels everything is discrete even distribution functions. There’s no sense in having a continuous spacial distribution sub Planck lengths.