I edited the question to explore definitions other than this question.
I`m trying to simplify
$$δ((x^2-a^2)^{1/2})$$
using
$$\delta\big(f(x)\big) = \sum_{i}\frac{\delta(x-a_{i})}{\left|{\frac{df}{dx}(a_{i})}\right|}$$
but the derivative in the denominator diverges in the points $a$ and $-a$. From what I've been reading here on the mathstackexchange, the argument f(x) of the Dirac delta must be continuously differentiable. The derivative of
$$(x^2-a^2)^{1/2}$$
is
$$\frac{x}{\sqrt{x^2-a^2}}$$ .
If the domain and image of f(x) are real, since, as far as I've learned, the Dirac delta argument cannot be complex, this derivative is discontinuous in the closed interval [-a;a]. Is this argument correct? From this, is it possible to say f(x) is not continuously differentiable and that the function $$δ((x^2-a^2)^{1/2})$$ is undefined?