Discontinuous function as a sum of step function and a continuous function

Question

If f:[a,b] is discontinuous at exactly one point then, is it possible to write the funtion as sum of a continuous function and a step function. If it was a jump or removable discontinuity then I believe it's true, I am not sure about the case of oscillatory discontinuity like sin(1/x).

Answer 1

If $f$ is a sum of a continuous function and a step function then it has right hand and left hand limits at every point. So any function which does not have right hand or left hand limits at a point cannot be written in this form. In particular, your example $\sin (\frac 1 x)$ works as a counter-example.