What is/are conditions for a function to be continuous on (a,b)?

The correct option is "Right continuous, left continuous, continuous at each point of (a,b)".

To understand why this is the case, let us break down the definition of continuity on an open interval $(a,b)$.

By definition, a real-valued function $f$ is continuous on an open interval $(a,b)$ if it is continuous at every point $c$ within that interval, meaning $a<c<b$.

For a function to be continuous at any given point $c\in (a,b)$, the following three conditions must be satisfied:
1. The function is defined at $c$, so $f\left(c\right)$ exists.
2. The limit of $f\left(x\right)$ as $x$ approaches $c$ exists.
3. The limit of the function as $x$ approaches $c$ is equal to the function's value at $c$.

Mathematically, the existence of the limit at $c$ requires both the left-hand limit and the right-hand limit to exist and be equal. This is expressed as:

$\underset{x\to c^{-}}{lim}f\left(x\right)=\underset{x\to c^{+}}{lim}f\left(x\right)=f\left(c\right)$

Here:
- The condition $\underset{x\to c^{-}}{lim}f\left(x\right)=f\left(c\right)$ defines left continuity at $c$.
- The condition $\underset{x\to c^{+}}{lim}f\left(x\right)=f\left(c\right)$ defines right continuity at $c$.

Therefore, for a function to be continuous at each point in the interval $(a,b)$, it must be both left continuous and right continuous at every such point, which establishes the equivalence and necessity of all three conditions combined.