Diferenciáveis

Diferenciáveis, also known as differentiable functions, are a fundamental concept in calculus, particularly in the study of real-valued functions of one or more real variables. A function is said to be differentiable at a point if it has a derivative at that point. This means that the function has a well-defined tangent line at that point, which is a line that best approximates the function near that point.

The concept of differentiability is closely related to that of continuity. A function that is differentiable at a point is necessarily continuous at that point. However, the converse is not true; a function can be continuous at a point but not differentiable there. For example, the absolute value function |x| is continuous at x = 0 but not differentiable there because the function has a "sharp corner" at this point.

Differentiability is a local property, meaning that a function is differentiable at a point if and only if it is differentiable in some open interval containing that point. However, a function can be differentiable at every point in its domain without being differentiable on the entire domain. For example, the function f(x) = |x| is differentiable on the intervals (-∞, 0) and (0, ∞), but not at x = 0.

The study of differentiable functions is crucial in many areas of mathematics and its applications, including optimization, numerical analysis, and differential equations. The concept of differentiability can be extended to functions of several variables, leading to the study of partial derivatives and the Jacobian matrix.