Differentiability and Continuity Differentiability A function is differentiable at a point if its derivative exists at that point. The derivative is...
Differentiability and Continuity
Differentiability
A function is differentiable at a point if its derivative exists at that point. The derivative is a measure of how the function's rate of change changes at that point.
Continuity
A function is continuous at a point if its limit at that point exists. The limit represents the value the function approaches as the input approaches that point.
Connection between Differentiability and Continuity
Differentiability and continuity are closely related concepts. A function is differentiable at a point if its derivative exists at that point, and the function is continuous at that point if its limit at that point exists.
Example
Consider the function (f(x) = x^2).
Differentiability: The derivative of (f(x)) is (2x).
Continuity: The limit of (f(x)) as (x) approaches (0) is (0), which exists, so (f(x)) is continuous at (x = 0).
Key Points
A function is differentiable at a point if its derivative exists at that point.
A function is continuous at a point if its limit at that point exists.
A function is differentiable if and only if it is continuous at all points in its domain