The definition of differentiability in mathematics is the property of a function being continuous and having a defined derivative at a given point.
To determine if a function is differentiable at a point, you can use the limit definition of a derivative. If the limit exists and is finite, then the function is differentiable at that point.
Differentiability and continuity are related concepts, but they are not the same. A function that is continuous at a point may or may not be differentiable at that point. Continuity requires the function to have a defined value at that point, while differentiability requires the function to have a defined slope at that point.
No, a function cannot be differentiable but not continuous. Differentiability implies continuity, so if a function is differentiable at a point, it is also continuous at that point.
Differentiability is a fundamental concept in calculus and is used in many areas of science and engineering. It is used to model and analyze rates of change, such as in physics, economics, and biology. It is also used in optimization problems, such as finding the maximum or minimum value of a function.