I'm dealing with functions of a special type whose derivatives are hard to compute, and which I'm trying to bound from below.
Let $A=[a,b)$ or any interval of your choice, open, closed or neither.
Let $f_k:A \to \mathbb{R}$ be a sequence of continuous (and continuously differentiable of some degree) functions converging (uniformly or pointwise, of your choice) to a function $g: A \to \mathbb{R}$. The function $g$ is continuous (and continuously differentiable of some degree) and monotone:
$x \leq y \implies g(x) \leq g(y)$ or $x \leq y \implies g(y) \leq g(x)$.
In addition, lets say that $f_k(a+), f_k(b-) < \infty$ and computable.
Under what conditions, if any, and involving no conditions imposed on the derivatives (other than their existence) of the functions $f_k$ of the sequence, can we say that there must be some $N > 0$ such that the functions $f_k$ of the sequence are monotone (with the same sign) whenever $k> N$. That is, is there an $N > 0$ such that the function $f_k$ is increasing (decreasing) whenever $k > N$?
In short, what can be said about the monotonicity of the functions $f_k$ given that they converge (in some sense) to a monotone function, if any?
There is hardly any hope to conclude anything about the monotonicity of the $f_k$ given some monotonicity of $g$.
Let $\Phi(x)$ be some on $[a,b]$ bounded function (e.g. a $C^\infty$-bump function, but any continuous and many more functions will work). Let $\Phi_k(x)=\Phi(x)/k$.
Then $\Phi_k\rightarrow 0$ pointwise, uniformly and in any $L^p$. The zerofunction is monotonous, but the $\Phi_k$ are (in general) not.