What is the mathematical expression for a function to be strictly increasing on (a,b)?

The correct option is:
$$x_1<x_2\Rightarrow f\left(x_1\right)<f\left(x_2\right)\forall x_1,x_2\in (a,b)$$

Understanding the Concept of a Strictly Increasing Function:
A function is said to be strictly increasing on an open interval $(a,b)$ if, as the input value increases, the corresponding output value also strictly increases. In other words, larger inputs must always yield strictly larger outputs.

Step-by-Step Breakdown of the Mathematical Expression:
1. Domain of Input Variables: We choose any two arbitrary points, $x_1$ and $x_2$, belonging to the open interval $(a,b)$. This is denoted by the universal quantifier symbol $\forall$ (which means "for all") and the membership symbol $\in$ (which means "belongs to" or "in"):
$$\forall x_1,x_2\in (a,b)$$
2. Input Relation: We establish an ordering between the two chosen points, assuming one is strictly smaller than the other:
$$x_1<x_2$$
3. Implication and Output Relation: For the function to be strictly increasing, this inequality relation must be preserved by the function. That is, the function value at the smaller point must be strictly less than the function value at the larger point:
$$f\left(x_1\right)<f\left(x_2\right)$$
The implication arrow $\Rightarrow$ (meaning "implies") connects these statements, showing that if the condition $x_1<x_2$ holds, it necessarily follows that $f\left(x_1\right)<f\left(x_2\right)$.

Why Other Options are Incorrect:
- The option containing $f\left(x_1\right)\ge f\left(x_2\right)$ defines a decreasing function.
- Options starting with $x_1=x_2$ are mathematically inconsistent for defining monotonic properties since equal inputs must produce equal outputs for any well-defined function ($x_1=x_2\Rightarrow f\left(x_1\right)=f\left(x_2\right)$).