In our course (Calc 1), our teachers write down a theorem that states that all elementary functions such as algebraic functions (polynomials, rational, irrational functions) and transcendental functions such as trigonometric and hyperbolic functions are continuous in their domains.
Strangely, a function such as $sin^x(x)$ defies this theorem (since it is discontinuous over the entire R), but our textbook does not bring this up.
The theorem is stated without a proof. My understanding is that this is confirmed by experience. The truth is to form is discontinuous functions, one has to recourse to piececwise defined functions, otherwise one has to wait for non-trivial examples of trignonometric series (Fourier series).
Is there a way to formally prove this theorem without recoursing to experience, even though it may seem obvious.