Différentiabilité

Différentiabilité is a fundamental concept in calculus and mathematical analysis that describes whether a function is "smooth" or has a "well-defined tangent line" at a given point. A function is differentiable at a point if its derivative exists at that point. The derivative of a function at a point represents the instantaneous rate of change of the function at that point, which geometrically corresponds to the slope of the tangent line to the function's graph at that point.

For a function f(x) of a single real variable, differentiability at a point 'a' means that the limit of the difference quotient, lim (h->0) [f(a+h) - f(a)] / h, exists. This limit is the definition of the derivative, denoted f'(a). If this limit does not exist, the function is not differentiable at 'a'. Common reasons for non-differentiability include sharp corners (like the absolute value function at x=0), cusps, vertical tangents, or discontinuities in the function.

In higher dimensions, for functions of multiple variables, differentiability is a more complex notion. A function is differentiable at a point if it can be locally approximated by a linear function. This involves the concept of partial derivatives and the Jacobian matrix. If a function is differentiable at a point, it is also continuous at that point. However, the converse is not always true; a function can be continuous at a point without being differentiable there. The concept of differentiability is crucial for many areas of mathematics, physics, engineering, and economics, underpinning methods like optimization, curve fitting, and the analysis of dynamic systems.