Continuity is a mathematical concept that deals with how a function behaves as we get close to a certain point. A function is a set of instructions that takes an input and gives an output. There are two types of continuity: point-wise continuity and uniform continuity.
Point-wise continuity: This means that as we get close to a certain point, the output of the function gets close to a certain value. Imagine that you are drawing a picture and you want to color in a shape. If you color in the shape carefully, the color will be smooth and there will not be any abrupt changes. This is an example of point-wise continuity.
Uniform continuity: This means that the function behaves smoothly over the whole range of values, not just at a single point. Imagine that you are drawing a big picture that covers an entire wall. If you color in the picture carefully, the color will be smooth and even across the whole wall. This is an example of uniform continuity.
In mathematics, understanding continuity is important because it helps us make predictions and understand how functions behave. It is used in many different fields, including science, engineering, and economics.