Differentiability

Differentiability is the property of a function having a well-defined derivative at a point or throughout a domain. It describes local behavior that can be approximated by a linear map.

Derivative: A derivative is the limit of a function’s difference quotient, measuring its instantaneous rate of change. Differentiability at a point means this limit exists there.
Limit of a function: A limit describes the value a function approaches as its input approaches a specified point. A derivative is defined through a limit of difference quotients.
Absolute value function: The absolute value function maps each real number to its nonnegative magnitude. Its corner at zero makes it continuous there but not differentiable.
Mean value theorem: The mean value theorem equates a function’s average slope on an interval with an instantaneous derivative inside it. Differentiability on the interval’s interior is one of its central hypotheses.
Newton's method: Newton’s method iteratively estimates a root using the function’s value and derivative. The method uses local linearization and can fail when derivatives are unavailable or poorly behaved.
Difference quotient: A difference quotient measures a function’s average rate of change across a nonzero input interval. Its limit as the interval shrinks determines whether a derivative exists.
Continuity: Continuity at a point means a function’s values approach its value at that point as inputs approach it. Differentiability implies continuity, but continuity alone does not imply differentiability.
Weierstrass function: The Weierstrass function is a continuous function that is nowhere differentiable. It shows that continuity across a whole domain does not guarantee a derivative anywhere.
Taylor's theorem: Taylor’s theorem approximates a sufficiently differentiable function by a polynomial and bounds the remainder. Higher derivatives strengthen local approximation beyond the first-order linear case.
Gradient descent: Gradient descent updates parameters opposite an objective function’s gradient to seek a minimum. Its standard update requires differentiability of the objective with respect to its parameters.

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