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If a function is differentiable at a point, then it is continuous at that point.
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.
A differentiable function (one variable) is a function that has a derivative at a point (meaning exists), and it is differentiable on if it is differentiable at every point of .
A differentiable map at a point is a map (with ) for which there exists a linear map such that
where is the Euclidean norm.
In this case is the Fréchet derivative of at , and (when it exists) it is represented in coordinates by the Jacobian matrix. Maps that are differentiable at every point of their domain are the basic objects of multivariable differentiability in higher dimensions, and higher smoothness is recorded by C^k maps.
A Fréchet derivative of a function (with ) at a point is a linear map such that
where is the Euclidean norm.
If such a map exists, it is unique; it is the linear part of the first-order approximation . When has partial derivatives, is represented by the Jacobian matrix at , and existence of is the defining condition for a differentiable map at .
A continuous map between topological spaces and is a function such that for every open set , the preimage is open in .
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.