I'm looking for a formula that, given a linear $x$ input, would yield values of $y$ in a "stairs" shape so to speak, in such a way that as the value of $x$ grows higher, the difference between each step is bigger, and it also takes longer to reach the next the step, as this graph (hopefully) illustrates:
As you have probably guessed by now I'm not a mathematician, but I'm fairly sure this has an easy solution. I've been playing around with modulus and powers but I couldn't quite get the graph above so far.
Edit: Graph updated.