Continuity and Differentiability

# Continuity and Differentiability

## Continuity and Differentiability Formulas

The continuity and differentiability formulas state that a function is continuous at a point if and only if its derivative exists at that point. A function is differentiable at a point if and only if its derivative exists and is continuous at that point.

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## What is Continuity?

Continuity refers to the uninterrupted flow of a substance or energy from one point to another. In physics, continuity is principle that states that physical laws are same in all directions and at all points in space. This principle is important in the study of fluids, which can flow in any direction.

## Formal Definition of Continuity

A function is continuous at a point if given any two points within the function’s domain, there exists a smooth curve that connects those points. In other words, a function is continuous at a point if the function can be drawn without any breaks or gaps.

## Geometrical Interpretation of Continuity

Function is continuous at point if given any two points within function’s domain, there exists smooth curve passing through those points. In other words, function is continuous at a point if it can drawn without any breaks or sudden changes in direction.

## Definition of Real Functions

Real function is mathematical relationship between two real variables, x and y, that assigns unique value to every point in plane. Graph of real function is a set of points, called points of intersection. In plane that corresponds to values of the function at different points in plane.

## What is Differentiability?

Differentiability is a measure of how much a function changes when its inputs change by a small amount. A differentiable function is one for which the derivative exists at every point in its domain.

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