How to prove that all elementary functions are continuous in their domain?

Question

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.

Answer 1

The definition:

The elementary functions (of $x$) include:

• Powers of $x$ : $x$, $x^2$, $x^3$, etc.
• Roots of x : $\sqrt{x}$, $\sqrt[3]{x}$, etc.
• Exponential functions: $e^{x}$
• Logarithms: $\log x$
• Trigonometric functions: $\sin x$, $\cos x$, etc.
• Inverse trigonometric functions: $\arcsin⁡ x$, $\arccos⁡ x$, etc.
• Hyperbolic functions: $\sinh x$, $\cosh x$, etc.
• All functions obtained by replacing $x$ with any of the previous functions
• All functions obtained by adding, subtracting, multiplying or dividing any of the previous functions

For the first seven entries on that list, we just verify that they're continuous on their domains.

Then we allow arithmetic operations... hey, there's a theorem we can quote: sums, products, negatives, and reciprocals of continuous functions are continuous, except for the case of a reciprocal at points where that function is zero (which we'll drop from the domain).

Then we allow composition, and again there's a theorem: the composition of continuous functions is continuous.

So there it is - start with the basics, and induct on the number* of arithmetic operations and compositions from there.

*This must be finite. Infinite sums allow for basically anything, which takes us outside the domain of elementary functions.