Question

How do you find the x values at which `f(x)=3x-cosx` is not continuous, which of the discontinuities are removable?

Answer 1

`f(x)=3x-cosx` is continuous `AA x in RR`, and so it does not have any discontinuities.

Explanation:

`3x` is continuous for all real numbers `x` (we write as `AA x in RR` which means for all `x` in the set of real numbers).

`cosx` is also continuous `AA x in RR`

Therefore any linear combination of the above is also continuous `AA x in RR`.

Hence `f(x)=3x-cosx` is continuous `AA x in RR`, and so it does not have any discontinuities.

You can see this visually by looking at the graph of `y)=3x-cosx`, which is essentially that of `y=3x` with a slight oscillation caused by the addition of `-cosx`

graph{3x-cosx [-30, 30, -30, 30]}