Differentiability is a mathematical concept that describes the smoothness of a function. A function is said to be differentiable at a point if the slope of the function at that point can be defined by a tangent line.
Differentiability is important because it allows us to analyze the behavior of a function at a specific point. It also enables us to find the rate of change or slope of a function at a given point, which is useful in many applications such as optimization, physics, and engineering.
The main condition for differentiability is that the limit of the difference quotient (also known as the derivative) exists as the distance between two points on the function approaches zero. In other words, the function must be continuous and have a defined slope at the point in question.
Yes, a function can be continuous but not differentiable. This can occur when there is a sharp turn or a corner in the function, where the slope changes suddenly and does not have a defined value at that point. An example of this is the absolute value function, which is continuous but not differentiable at x=0.
Differentiability is closely related to the concept of limits. In fact, the derivative of a function at a point is defined as the limit of the difference quotient as the distance between two points approaches zero. Differentiability is essentially the property of a function that allows us to take the limit of the difference quotient and obtain a well-defined value.