Questions about $g(x)= \inf_{h \in \mathbb{R}} \frac{f(x+h)-f(x)}{h}$. I have several question about the following functions
\begin{align}
g_1(x)= \inf_{h \in \mathbb{R}} \frac{f(x+h)-f(x)}{h} \\
g_2(x)= \inf_{h \in \mathbb{R}} \left| \frac{f(x+h)-f(x)}{h} \right|
\end{align}
Note that if $f$ is differentiable at $x$ then
\begin{align}
g_1(x) \le f^\prime(x). 
\end{align}


*

*Do $g_1(x)$ and $g_2(x)$  have names?

*Do $g_1(x)$ and $g_2(x)$ have any applications?

*What is required for $g_1(x)=f^\prime(x)$  for some given $x$?

*Are there better bounds on $g_1(x)$ than  $g_1(x) \le f^\prime(x)$?

 A: I am not completely sure about your first two questions, but when it comes to question three, you have the following condition:
Given some fixed $x_0 \in \mathbb{R}$, we have that:


*

*$g_1(x_0) = f'(x_0) \iff \inf_{h \in \mathbb{R}} \frac{f(x_0+h) - f(x_0)}{h} = f'(x_0)$

*$g_1(x_0) = f'(x_0) \iff \forall h \in \mathbb{R}, \frac{f(x_0+h) - f(x_0)}{h} \geq f'(x_0)$

*$g_1(x_0) = f'(x_0) \iff \forall h \in \mathbb{R}, f(x_0+h) \geq f(x_0) + f'(x_0)h$

*$g_1(x_0) = f'(x_0) \iff \forall x \in \mathbb{R}, f(x) \geq f(x_0) + f'(x_0)(h-x)$


You can best see this with a picture, but $f$ being differentiable at $x_0$ is equivalent to there being a best fit linear equation for $f$ around $x_0$. Incidentally this best fit linear equation is always
$L_{f,x_0} \colon \mathbb{R} \to \mathbb{R}$ given by $L_{f,x_0}(x) = f(x_0) + f'(x_0) (h-x)$
So in other words the equality $g_1(x_0) = f'(x_0)$ holds if and only if the best fit linear equation $L_{f,x_0}$ forms a lower bound for $f$ (i.e. $f \geq L_{f,x_0}$).

Another way of thinking about this would be graphically. If you were to plot the graphs of $y = f(x)$ and $y = L_{f,x_0}(x)$, then equality will hold if and only if the graph of $f$ remains above (or touching) $L_{f,x_0}$ at all times. See the example below:

As you should be able to see in the above example - the equality $g_1(x) = f'(x)$ will hold for all $x \in \mathbb{R}$, because $f$ will always be above the best linear fit, no matter which point you chose to take it.

As for your forth question - for general differentiable functions, there will never be a better bound than $g_1(x) \leq f'(x)$. Taking the function $f_c(x) = cx$ gives a simple example where $g(x) = f_c'(x)$ for all $x \in \mathbb{R}$. Therefore $g_1$ can not be lower than $f'$ for general differentiable functions.

EDIT: I should also emphasize that $L_{f,x_0}$ being a lower bound for $f$ is the ONLY requirement for $g(x_0) = f'(x_0)$. Take this very crude Microsoft Paint drawing as an example of a crazy function that still obeys the equality:


FURTHER EDIT: Also whilst I don't think there are any names for the specific functions you gave, they are similar to "limit superior" and "limit inferior" functions. They have a practical application in that they always exist (even if a function is nowhere differentiable), and if:
$\inf_{h \in (x_0-r, x_0+r)} \frac{f(x_0 + h) - f(x_0)}{h} \sim \sup_{h \in (x_0-r, x_0+r)} \frac{f(x_0 + h) - f(x_0)}{h}$ as $r \to 0$
then this proves that $f$ is differentiable at $x_0$, with $f'(x_0) = \lim_{r \to 0} \inf_{h \in (x_0-r, x_0+r)} \frac{f(x_0 + h) - f(x_0)}{h}$.
