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The limit of a function's difference quotient at an interior point.
A function , defined on an open interval , is differentiable at if the finite limit
exists. The number is the derivative of at .
This is a special instance of a limit at a point applied to the difference quotient. Existence of the derivative is the basic notion behind differentiability in one variable, and it implies continuity via differentiability implies continuity.
A limit at a point for a function (with ) at a point is a number such that: for every there exists with the property that whenever and , one has . Typically one assumes that is a limit point of .
The inequalities use the absolute value to measure distance on the real line. One-sided variants are captured by the one-sided limit.
A function is differentiable at if the finite limit
exists. It is differentiable on if it is differentiable at every point of , with one-sided limits used at endpoints of an interval.
Differentiability implies continuity: Let be a function on an interval , and let be an interior point. If is differentiable at , then
so is continuous at .
More generally, if is differentiable at in the sense of the Fréchet derivative, then is continuous at (i.e. a continuous map at that point). This connects derivatives to limits at a point.