Total differentiability and linear approximation

Let f map an open subset of Euclidean n-space into Euclidean m-space, and let a be an interior point. Its derivative at a is a linear map L for which subtracting f(a)+Lh leaves a remainder whose norm divided by the norm of h tends to zero. Both the direction and length of h may change during that limit. The definition is consequently stronger than checking a finite list of lines. The derivative is unique: evaluating two proposed derivatives on displacements tv forces their difference to vanish on every fixed vector v.

A differentiable function is continuous at the point because the linear part tends to zero and the remainder is smaller still. In coordinates, applying the derivative to each coordinate basis vector recovers the corresponding partial derivative. Thus the matrix of a derivative, when it exists, must be the Jacobian matrix. Computing that candidate is useful, but it does not by itself prove that the required remainder estimate holds. A proof must control arbitrary nearby displacements, either directly or through a theorem whose hypotheses have been checked.

Continuous first partial derivatives on a neighborhood imply total differentiability there. To understand the condition, move between two nearby points one coordinate at a time and apply the one-variable mean value theorem on each segment. Continuity makes the intermediate partial derivatives close to their values at the base point, so the accumulated error is small relative to displacement. Continuity of the partials is sufficient rather than necessary: a differentiable map can have a derivative that fails to vary continuously.

There are even functions with every directional derivative at the origin but no total derivative there. Define f(x,y)=x cubed divided by x squared plus y squared away from the origin, and define its origin value as zero. Along a displacement tv, the directional derivative is v₁ cubed divided by v₁ squared plus v₂ squared for nonzero v. This dependence on v is not linear. The coordinate partials suggest the map L(h)=h₁, but on h=(t,t), f(h)=t/2 and the normalized error is a nonzero constant. The candidate therefore fails.

A derivative predicts changes, not absolute values without a base value. For a scalar output, its row matrix acts by a dot product with the gradient. In a calibrated physical model, each column carries the output units divided by the associated input units. Comparing coordinate magnitudes may require nondimensionalizing the variables; a norm that mixes meters with seconds without a declared scale hides that choice. The existence of a derivative is unchanged by fixed invertible linear rescaling, although numerical condition estimates do change.

The little-o remainder is an asymptotic promise, not a universal numerical tolerance. To certify an error at a specified displacement, derive a concrete bound, perhaps using a Hessian bound on a neighborhood. Nor does an accurate local approximation establish global invertibility or behavior far from the base point. A useful workflow separates three tasks: compute the candidate derivative, justify differentiability, and estimate the residual over the intended operating range. That separation prevents an attractive tangent picture from being mistaken for a global model.