Let $f : (a,b) \rightarrow \mathbb{R}$ denote a continuous function. Suppose $x \in (a,b)$ satisfies $f(x)=0$. Then there exists $r>0$ such that:
The restriction of $f$ to $[x,x+r)$ is either consistently $\geq 0$, consistently $=0$, or consistently $\leq 0$.
The restriction of $f$ to $(x-r,x]$ is either consistently $\geq 0$, consistently $=0$, or consistently $\leq 0$.
Now in practice, we're often only interested in those zeros such that the answer is different on the left and right. For instance, if $f(x)$ is negative on the left of $x$ and positive on the right, that's an important zero of $f$. But if $f$ is negative on both the left and the right of $x$, there are situations where this zero can safely be ignored. For example, if we're looking for the turning points of $f(x)$, some of the zeroes of $f'(x)$ can be ignored, because they correspond to points of inflection of $f(x)$ and hence aren't relevant to the problem. If you think about it for a bit, you'll see that it's really the roots of $f'(x)$ that are "non-trivial", in the sense of having different behavior on each side, that matter with respect to the problem of minimizing and maximizing $f(x)$ on an interval.
My question, quite simply, is:
Question. For continuous $f:(a,b) \rightarrow \mathbb{R}$, is there a name for those zeroes of $f$ such that it behaves differently on each side of that point?
I got dragged here by wondering who asked a different question, and then looking at some of @goblin's other questions.
I just want to note that the first statement about $f$ is false, which makes the rest of the question sort of moot, although @AntonioVargas's suggestion of "weakly changes sign" is a good term for something related to @goblin's original question.
To see that the first statement of the question is false, look at the function
$$ f(x) = \begin{cases} 0 & x = 0 \\ x^2sin(\frac{1}{x}) & x \ne 0 \end{cases}. $$
It is differentiable at $0$ (with derivative $0$ there), but for every $r > 0$, there are points of the interval $[0, r)$ where $f$ is positive, negative, and zero. (To find them, look for the first integer multiple of $\frac{1}{2\pi}$ that's larger than $1/r$. Call this $u$. Then $f(u) = 0$, and $f(u^{+}) < 0$ and $f(u^{-}) > 0$, where $u^{\pm}$ denote any small-enough perturbations of $u$.)