Why is the direct substitution property so specific

Question

Mt text book states the Direct Substitution Property as

If f is a polynomial or a rational function and a is in the domain, then $$\begin{align*} \lim_{x\to a} f(x)=f(a) \end{align*}$$ Why does this definition not include radical functions?

Answer 1

First, that's a theorem, not a definition. (It's not defining the limit, just stating that it has a certain value under those hypotheses.)

Anyway, it's true in much more generality: If $f$ is any continuous function on the real numbers (or a subset of the real numbers), then for all $a$ in the domain of $f$, $$\lim_{x \to a} f(x) = f(a).$$ The converse is also true: if a function $f$ satisfies the above condition, then it is continuous.

So, what that theorem in your book really means is that polynomials and rational functions are continuous on their domain. (Rational functions may have poles at which the function isn't defined, but those aren't in the domain, so the function is still continuous.)

I don't know why your book states it like that; it seems very misleading, since it suggests (incorrectly) that those types of functions are the only ones with that property.

Also, if by "radical function", you're just talking about the non-negative real-valued $n$-th root, then it is indeed continuous as a function from $\mathbb{R}_{\geq 0}$ to $\mathbb{R}_{\geq 0}$, where $\mathbb{R}_{\geq 0}$ denotes the set of non-negative real numbers. (If you want to consider the full, complex-valued $n$-th root function, on the other hand, things get more complicated with multivalued functions and branch cuts, but that's probably beyond the scope of your course.)