"Real World" Applications of an Efficient Root Finder
Sure.  If this works for any function $f$ that can be computed efficiently, such an algorithm would let you solve any NP problem.

Let $g:\{0,1\}^n \to \{0,1\}$ be an arbitrary function that is efficiently computable.  When $x$ is an integer, define $f(x) = g(x_1,\dots,x_n)$ where $x_1\cdots x_n$ is the binary representation of $x$; when $x$ is not an integer, define $f(x)$ to linearly interpolate between $f(\lfloor x \rfloor)$ and $f(\lceil x \rceil)$, i.e.,

$$f(x) = f(\lfloor x \rfloor) + (f(\lceil x \rceil) - f(\lfloor x \rfloor))(x - \lfloor x \rfloor).$$

Then if we apply your algorithm to $f$, we find an efficient way to compute $x_1,\dots,x_n$ such that $g(x_1,\dots,x_n)=0$.  This in turn lets you efficiently solve any NP problem.

For instance, you could break the RSA cryptosystem efficiently, you could mint Bitcoin coins efficiently, you could solve optimization and operations research problems efficiently, and so on.



If this is restricted to only $f$ that are low degree polynomials, we already know of algorithms to solve the problem, so it likely would not represent an advance over the state of the art and likely wouldn't have much of an impact in practice, unless your method was significantly more efficient than existing methods.