Continuity refers to the property that a function has when there are no abrupt changes or breaks in its behavior. A function is considered continuous if it does not experience any sudden jumps, holes, or gaps.

Point Continuity: A function is said to be point continuous if it is continuous at every individual point within its domain. This means that the function must have a well-defined value at each point and its limit exists at that point.

Interval Continuity: A function is interval continuous if it is continuous over an entire interval or range of values. This implies that the function is continuous at every point within that interval.

Limits: The concept of limits is closely associated with continuity. A function is continuous at a point if and only if its limit exists at that point and is equal to the value of the function at that point. Limits allow us to describe the behavior of a function as it approaches a certain point.

Jumps and Discontinuities: A jump in a function occurs when there is an abrupt change in its value, causing a discontinuity. Discontinuities can be classified into different types such as removable, jump, and essential discontinuities.

Smoothness: A smooth function is one that does not have any sudden changes in its slope or curvature. It is continuous and has well-behaved derivatives, meaning that the rate of change of the function is consistent and predictable.

Continuity plays a crucial role in various mathematical disciplines and real-world applications:

• In calculus, continuity is fundamental for the study of limits, derivatives, and integrals.
• In physics, continuity is applied to describe the continuous flow of fluids, conservation laws, and the behavior of waves.
• In computer science, continuity is important for smooth animations, algorithms, and numerical methods.
• In economics and finance, continuity is relevant when analyzing continuous functions such as demand and supply curves.