Differentiability

Differentiability is a fundamental concept in calculus that describes the property of a function being smooth enough to have a well-defined derivative at a given point. A function is said to be differentiable at a point if the limit of its average rate of change, as the interval approaches zero, exists and is finite. This derivative represents the slope of the tangent line to the function's graph at that point.

Mathematically, a function \(f(x)\) is differentiable at \(x=a\) if the following limit exists: \\

\[

f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}

\]

If this limit exists, \(f\) is differentiable at \(a\), and \(f'(a)\) is the derivative. Differentiability implies continuity, meaning that a function that is differentiable at a point must also be continuous there. However, the converse is not true, as some continuous functions are not differentiable (e.g., functions with sharp corners or cusps).

Differentiability is essential in various fields, including physics, engineering, and economics, to analyze rates of change, optimize functions, and model real-world phenomena. Functions can be differentiable over an interval, a neighborhood, or their entire domain, depending on their smoothness. The study of differentiability involves exploring concepts like differentiability classes, higher derivatives, and the relationship between differentiability and other properties such as convexity and analyticity.

Discontinuities, corners, cusps, and vertical tangents are common features where differentiability fails. Understanding where and why a function is differentiable allows mathematicians and scientists to better analyze its behavior and apply calculus techniques effectively.