My professor has this exercise in his notes. I may be able to prove it but I don't know what he pseudo-sine function is.
Let be the pseudo-sine function. Let for . Prove has a removable discontinuity at .
The pseudo-sine function might be , for .
In this case, we have for , and it is not hard to check that this function has a removable discontinuity at .
Do you know how to prove this?
Also, you better ask your professor, if this is the right definition of the pseudo-sine function (I recall having seen that name somewhere, definitely involving the function , but it might have been or something similar).
A more general concept of "pseudo-sine" is any function that is (quas-) periodic and bounded. Of course, for this proof, you only need "bounded". If then . Then, as x goes to 0, so the limit exists and is 0. You only need "exists" to prove there is a removable discontinuity at x= 0.