# Non-integrable bounded function which has antiderivatives

I would like to construct a bounded, real function $$f:[a,b] \to \mathbb{R}$$, which is not Riemann integrable, but has antiderivatives. I can easily construct an unbounded function with this property, but finding a bounded one is significantly more difficult.
All I could find was this topic (Non-integrable function that has an antiderivative) which offers some valuable insight into this matter, but doesn't actually construct a function with this property.
I also wonder if such a function can be expressed in terms of elementary functions because in the topic above some results from measure theory are mentioned and these are beyond my reach.