Core concept in calculus

Differentiability is a fundamental concept in calculus that relates to the smoothness of a function's graph and the existence of its derivative at each point in its domain

Existence of derivative

Derivative as instantaneous rate of change

A function is considered differentiable at a point if its derivative, representing the instantaneous rate of change, exists at that point

Linked to local behavior and graphical smoothness

Differentiability is closely tied to a function's local behavior and the smoothness of its graph

Assessment of differentiability

Differentiability is evaluated by examining a function's limits and continuity, which are essential tools in calculus for analyzing its behavior near specific points

Foundation for defining continuity and derivatives

Limits provide the basis for defining both continuity and derivatives in calculus

Definition of limits

A limit captures the value that a function approaches as its input nears a certain point

Requirements for continuity

For a function to be continuous at a point, it must be defined at that point, its limit as the input approaches that point must exist, and this limit must equal the function's value at that point

Fundamental theorem in calculus

The fundamental theorem states that differentiability implies continuity, meaning that if a function is differentiable at a point, it is also continuous at that point

Proof of the theorem

The proof involves showing that the limit of the difference quotient, which defines the derivative, is finite as the input approaches the point of interest

Non-differentiability

Causes of non-differentiability

Non-differentiability can occur at points of sharp corners, cusps, or vertical tangents, as well as any form of discontinuity

Graphical analysis

Graphical analysis can help determine if a function is differentiable, with functions exhibiting sharp corners, cusps, vertical tangents, or discontinuities being likely non-differentiable

Evaluating continuity and differentiability

In practical applications, continuity and differentiability are often evaluated by examining a function piecewise and computing limits

Piecewise-defined functions

A piecewise-defined function may be continuous at a point if the one-sided limits from either side of the point are equal, but for differentiability, the left-hand and right-hand derivatives must also be equal

Mismatch in derivatives

A mismatch in derivatives, as seen in functions with abrupt changes in direction or kinks, indicates non-differentiability despite the function being continuous