In the realm of mathematics, a closed interval is any set of numbers that is bounded. A set of numbers is bounded when its values are limited by two distinct numbers, which are known as the upper and lower bounds. The function f is differentiable on the closed interval if it is continuous and its derivative exists over the entire interval.

The concept of differentiability is essential to calculus and is used to measure the rate at which a function changes. An important property of differentiability is that it allows us to compute the area beneath a curve, as well as to find the instantaneous rate of change of a function.

In order to be differentiable on the closed interval, a function must be both continuous and have a unique derivative value for every point on the interval. A function is continuous when its graph is continuous and it does not contain any breaks or jumps.

In addition, the derivative of a function must exist for every point on the interval. The derivative is the rate of change of the function at a particular point, and is calculated by taking the limit of the function as x approaches the point. If the derivative does not exist for a point, then it is not differentiable at that point.

The concept of differentiability can be used to analyze functions and determine properties such as the area under a curve, or the instantaneous rate of change. Differentiability is an essential part of calculus and is used to study functions on the closed interval.

The function f is differentiable on the closed interval if it is continuous and its derivative exists over the entire interval. In order to be differentiable on the closed interval, a function must be both continuous and have a unique derivative value for every point on the interval. Differentiability is an important concept in calculus and can be used to analyze functions and determine properties such as the area under a curve, or the instantaneous rate of change.

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