What is the mathematical expression for f is right continuous on (a,b)?

The correct option is: lim_x→a⁺f(x) = f(a).

Understanding Continuity:
A function is continuous at a point if the limit of the function as it approaches that point is equal to the value of the function at that point. However, when we look at the boundaries of an interval, we often discuss one-sided continuity.

What is Right Continuity?
Right continuity (or continuity from the right) means that as we approach a point from values greater than that point (from the right-hand side on a number line), the limit of the function matches the function's value at that point.

For a point
$a$
, the mathematical definition of right continuity of a function
$f$
is expressed as:

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

where the superscript "+" on the limit denotes that
$x$
approaches
$a$
from the right (i.e.,
$x>a$
).

Evaluating the Options:

Option 1:
$\underset{x\to a^{+}}{lim}f\left(x\right)=f\left(a\right)$
correctly defines right continuity at the left endpoint
$a$
.

Option 2:
$\underset{x\to a^{+}}{lim}f\left(x\right)=f\left(b\right)$
is incorrect because the limit as
$x$
approaches
$a$
is compared to the function value at
$b$
instead of
$a$
.

Option 3:
$\underset{x\to b^{-}}{lim}f\left(x\right)=f\left(a\right)$
is incorrect because it mixes the left-hand limit at
$b$
with the function value at
$a$
.

Option 4:
$\underset{x\to a^{-}}{lim}f\left(x\right)=f\left(a\right)$
defines left-continuity at
$a$
, not right-continuity.