What is the mathematical expression for the definition of continuity?

The correct option is: lim_x→cf(x) = f(c) ∀ c ∈ (a,b).

To understand why this is the mathematical expression for the definition of continuity on an interval, we can break it down into two main concepts: continuity at a single point, and continuity over a set of points.

1. Continuity at a Point
A function $f\left(x\right)$ is continuous at a specific point $x=c$ if the value of the function at that point matches the limit of the function as $x$ approaches $c$. Mathematically, this is written as:

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

For this condition to hold, three underlying requirements must be met:
• The function must be defined at $c$ (so $f\left(c\right)$ is a real number).
• The limit of the function as $x$ approaches $c$ must exist.
• The value of this limit must equal the functional value $f\left(c\right)$.

2. Extending Continuity to an Interval
A function is defined as continuous on an open interval $(a,b)$ if it is continuous at every single point inside that interval.
To express this requirement mathematically, we use logic symbols:
• The symbol $\forall$ represents "for all" or "for every".
• The symbol $\in$ represents "belongs to" or "is an element of".

Therefore, we require the limit equation to hold for every point $c$ that belongs to the interval $(a,b)$:

$$\forall c\in (a,b)$$

Combining the limit equation with the interval statement gives us the full, rigorous definition of continuity on the interval $(a,b)$:

$$\underset{x\to c}{lim}f\left(x\right)=f\left(c\right)\forall c\in (a,b)$$