Reference

The epsilon-delta definition of continuity is a rigorous way of defining when a function is continuous at a particular point. According to this definition, a function ( f(x) ) is continuous at a point ( x = a ) if for every positive number ( \varepsilon ) (no matter how small), there exists a positive number ( \delta ) such that whenever the distance between ( x ) and ( a ) is less than ( \delta ) (i.e., ( |x - a| < \delta )), the distance between ( f(x) ) and ( f(a) ) is less than ( \varepsilon ) (i.e., ( |f(x) - f(a)| < \varepsilon )).

This can be understood as ensuring that small changes in ( x ) around ( a ) result in small changes in the value of the function.

For a more detailed explanation, you can refer to this article on Stack Exchange.

Here is also a helpful video explanation of the epsilon-delta definition of continuity.