Differentiability at a Point: A Rigorous Perspective

Behind this mask there is more than just flesh. Beneath this mask there is an idea… and ideas are bulletproof, Alan Moore

Recall

A complex number is specified by an ordered pair of real numbers (a, b) ∈ ℝ² and expressed or written in the form z = a + bi, where a and b are real numbers, and i is the imaginary unit, defined by the property i² = −1 ⇔ i = $\sqrt{-1}$, e.g., 2 + 5i, $7\pi + i\sqrt{2}.$ ℂ= { a + bi ∣a, b ∈ ℝ}.

Definition. Let D ⊆ ℂ be a set of complex numbers. A complex-valued function f of a complex variable, defined on D, is a rule that assigns to each complex number z belonging to the set D a unique complex number w, f: D ➞ ℂ.

• The set D is called the domain of the complex function f, D = Dom(f).
• The set of all actual outputs {f(z): z ∈ D} is called the range (or image) of the complex function f, also denoted f(D) = {f(z): z ∈ D}.

We often call the elements of D as points. If z = x+ iy ∈ D, then f(z) is called the image of the point z under f. f: D ➞ ℂ means that f is a complex function with domain D. We often write f(z) = u(x ,y) + iv(x, y), where u, v: ℝ² → ℝ are the real and imaginary parts.

Definition. Let D ⊆ ℂ, $f: D \rarr \Complex$ be a function and z₀ be a limit point of D (so arbitrarily close points of D lie around z₀, though possibly z₀ ∉ D). A complex number L is said to be a limit of the function f as z approaches z₀, written or expressed as $\lim_{z \to z_0} f(z)=L$, if for every epsilon ε > 0, there exist a corresponding delta δ > 0 such that |f(z) -L| < ε whenever z ∈ D and 0 < |z - z₀| < δ.

Why 0 < |z - z₀|? We exclude z = z₀ itself because the limit cares about values near z₀, not at z₀ itself. When z₀ ∉ D, you cannot evaluate f(z₀), so you only care about z approaching z₀. When z₀ ∈ D, you still want the function’s nearby behavior; this separates “limit” from “value.”

Equivalently, if ∀ε >0, ∃ δ > 0: (for every ε > 0, there exist a corresponding δ > 0) such that whenever z ∈ D ∩ B'(z₀; δ), f(z) ∈ B(L; ε) ↭ f(D ∩ B'(z₀; δ)) ⊂ B(L; ε).

If no such L exists, then we say that f(z) does not have a limit as z approaches z₀. This is exactly the same ε–δ formulation we know from real calculus, but now z and L live in the complex plane ℂ, and neighborhoods are round disks rather than intervals.

Continuity in the complex plane

Definition. Let D ⊆ ℂ. A function f: D → ℂ is said to be continuous at a point z₀ ∈ D if given any arbitrarily small ε > 0, there is a corresponding δ > 0 such that |f(z) - f(z₀)| < ε whenever z ∈ D and |z - z₀| < δ.

In words, arbitrarily small output‐changes ε can be guaranteed by restricting z to lie in a sufficiently small disk of radius δ around z₀.

Alternative (Sequential) Definition. Let D ⊆ ℂ. A function f: D → ℂ is said to be continuous at a point z₀ ∈ D if for every sequence {z_n}^∞_n=1 such that z_n ∈ D ∀n∈ℕ & z_n → z₀, we have $\lim_{z_n \to z_0} f(z_n) = f(z_0)$ .

Global Continuity

Definition. A function f: D → ℂ is said to be continuous if it is continuos at every point in its domain (∀z₀ ∈ D).

Differentiability at a point

Differentiability in higher dimensions is far richer than the single‐variable “limit of difference quotient.” It asks: can a function f : ℝⁿ → ℝ^m be locally approximated by a linear map? The answer is encoded in the Jacobian matrix Df(x), which —if it exists— provides the unique best affine approximation f(x+h) ≈ f(x) + Df(x)h.

Definition. Differentiability at a point. Let $f : ℝ^n \to ℝ^m$ be a function and let x be an interior point of the domain of f, $x \in \text{interior(dom f)} $. The function f is differentiable at x if there exists a matrix $Df(x) \in ℝ^{m \times n}$ that satisfies $\lim_{\substack{z \in \text{dom} f \\ z \neq x, z \to x}} \frac{||f(z) - f(x) - Df(x)(z-x)||_2}{||(z-x)||_2} = 0$ [*]

I should stress why x∈int(domf) is needed, that is because we require that for small h, x + h stays in the domain.

This matrix Df(x) is called the derivative or the Jacobian matrix of f at the point x.

Key Points

1. The condition $x \in \text{interior(dom f)}$ is very important because it ensures that for small displacements or perturbations z - x (or $h \in ℝ^n$ in the alternative notation), the point z (or x + h) remains within the domain of f. This is critical for the limit to be well-defined.
2. $z \neq x, z \to x$, we are calculating the limit, meaning that z approaches x, but z is never actually equal to x because we are looking at the rate of change as we get arbitrary close to x, not the value at x itself. It emphasizes that the behavior of f near x, not exactly at x, determines differentiability.
3. Uniqueness of the Jacobian. If the limit exists (equivalently, if such a matrix Df(x) satisfying the limit in [*] exits), the Jacobian is unique. In other words, there’s only one linear transformation that satisfies the definition and best approximates the function f at the point x (first order).
4. Df(x)(z -x) represents the best linear approximation to the change in f near x. Df(x) is the Jacobian matrix (which represents the derivative as a linear transformation) and it’s multiplied by the vector (z - x) (which is a vector in ℝⁿ). It tells you how f changes in direction and magnitude when moving a small amount from x in any direction.
5. Why the Norm Appears. The Euclidean norm ||···||₂ denotes the Euclidean norm of a vector that measures the length or magnitude of the vector. This is essential because we are dealing with vectors (not real numbers) in ℝⁿ and ℝ^m and we need a way to measure their size or magnitude (we are working in vector spaces).
6. The limit equal to 0: The entire expression inside the limit $\frac{||f(z) - f(x) - Df(x)(z-x)||_2}{||(z-x)||_2}$ represents the relative error between the true change (f(z) - f(x)) and the linear approximation. The limit being 0 means that this relative error becomes arbitrarily small as z gets closer and closer to x. In other words, the linear approximation becomes increasingly accurate as we zoom in on the point x.
7. Alternative Formulation using h. If we write or substitute z = x + h (h = z - x), then as z → x, h → 0. This gives us an alternative or equivalent form: $\lim_{\substack{x + h \in dom f \\ h \neq 0, h \to 0}} \frac{||f(x+h) - f(x) - Df(x)h||_2}{||h||_2} = 0$. This version is especially useful when writing Taylor expansions or computing limits directly.
8. How to Compute Df(x) The Jacobian matrix is formed by taking partial derivatives of each component function: $Df(x) = \frac{∂f_i(x)}{∂x_j}_{1≤i≤m,1≤j≤n}$. This means the entry in the i-th row and j-th column of Df(x) is the partial derivative of the i-th component function fᵢ with respect to the j-th variable xⱼ, evaluated at the point x. More explicitly, the Jacobian is:

Df(x) = $\Biggl (\begin{smallmatrix}\frac{∂f_1(x)}{∂x_1} & \frac{∂f_1(x)}{∂x_2} & ··· & \frac{∂f_1(x)}{∂x_n}\\\\ \frac{∂f_2(x)}{∂x_1} & \frac{∂f_2(x)}{∂x_2} & ··· & \frac{∂f_2(x)}{∂x_n}\\\\· & · & · & ·\\\\\\\\· & · & · & ·\\\\\\\\· & · & · & ·\\\\\frac{∂f_m(x)}{∂x_1} & \frac{∂f_m(x)}{∂x_2} & ··· & \frac{∂f_m(x)}{∂x_n}\end{smallmatrix}\Biggr )$. The Jacobian Df(x) is an m x n real matrix and this is the practical way to compute the Jacobian matrix. This matrix represents the linear map that best approximates f near x.

The definition of differentiability generalizes the familiar derivative from single-variable calculus. Instead of a number, the derivative becomes a matrix — the Jacobian — which represents the best linear approximation to the function at a point.

It captures the idea that a function can be locally approximated by a linear transformation. The Jacobian matrix is the matrix representation of this linear transformation, and its entries are the partial derivatives of the component functions. The use of norms is crucial for making the definition rigorous in higher dimensions. The condition x ∈ interior dom f ensures that we can consider small perturbations around x within the function’s domain. The use of limits guarantees that this approximation becomes arbitrarily good as we zoom in on the point of interest.

Examples

• Jacobian of the identity function. Let f: $ℝ^n \to ℝ^m$ be a function defined by $f(\vec{x}) = \vec{x}$. This function simply returns its input. Each component function is $f_i(\vec{x}) = x_i$. The partial derivative of $f_i(\vec{x})$ with respect to x_j is: $Df(\vec{x})_{ij} = \frac{∂f_i(\vec{x})}{∂x_j} = \delta(i, j)$ where δ(i, j) is the Kronecker delta, defined as: δ(i, j) = 1 if i = j, δ(i, j) = 0 if i ≠ j. This is precisely the definition of the n × n identity matrix, Df(x) = I_n.

• Jacobian of a linear transformation. Let f: $ℝ^n \to ℝ^m$ be a function defined by $f(\vec{x}) = A\vec{x}$ where A = {a_ij} ∈ ℝ^{m x n} is a constant m x n real matrix. Then, the i-th component function of f is given by: $f_i(\vec{x}) = \sum_{j=1}^n a_{ij}x_j$. Thus, the partial derivative of $f_i(\vec{x})$ with respect to x_j is: $Df(\vec{x})_{ij} = \frac{∂f_i(\vec{x})}{∂x_j}$ = a_ij. Thus, the Jacobian matrix $Df(\vec{x})$ is a m x n matrix where the (i, j)-th entry is a_ij. Conclusion: The Jacobian matrix is just the matrix A itself: $Df(\vec{x}) = A$.

Example (m = 2, n = 3). If A = $(\begin{smallmatrix}a_{11} & a_{12} & a_{13}\\\\a_{21} & a_{22} & a_{23}\end{smallmatrix})$ and $f(\vec{x}) = A\vec{x}$, then $Df(\vec{x}) = A = (\begin{smallmatrix}a_{11} & a_{12} & a_{13}\\\\a_{21} & a_{22} & a_{23}\end{smallmatrix})$

• Jacobian of an Affine Transformation. Let f: $ℝ^n \to ℝ^m$ be a function defined as $f(\vec{x}) = A\vec{x} + b$ where A = {a_ij} ∈ ℝ^{m x n} (a m x n real matrix), b ∈ ℝ^m (a constant vector). Then, the i-th component of the function f is: $f_i(\vec{x}) = \sum_{j=1}^n a_{ij}x_j + b_i$. The partial derivative of $f_i(\vec{x})$ with respect to x_j is: $Df(\vec{x})_{ij} = \frac{∂f_i(\vec{x})}{∂x_j}$ = a_ij (the constant term b_i disappears when we take the derivative). Thus, the Jacobian matrix $Df(\vec{x})$ is a m x n matrix where the (i, j)-th entry is a_ij. This is again the matrix A, $Df(\vec{x}) = A$. This shows that adding a constant vector does not change the Jacobian, as constant translations do not affect the local linear behavior.

• Jacobian of a Simple Nonlinear Function. Let f: $ℝ^2 \to ℝ^2$ (m = n = 2) be a function defined by f(x, y) = (x² + y, xy), then the Jacobian matrix is: $Df(\vec{x}) = (\begin{smallmatrix}2x & 1\\\\ y & x\end{smallmatrix})$

• Jacobian of a Vector-Valued Function from $ℝ^2 \to ℝ^3$. Let f(x, y) = $(\begin{smallmatrix}x + y\\\\ x^2y\\\\ sin(xy)\end{smallmatrix}), f: ℝ^2 \to ℝ^3$. So the Jacobian is: $Df(x, y) = (\begin{smallmatrix}1 & 1\\\\ 2xy & x^2\\\\ ycos(xy) & xcos(xy)\end{smallmatrix})$

• Quadratic and Squared Norm: ∣∣x∣∣²

Quick Jacobian Recipes

Differentiable function

Definition. A function f is called differentiable if its domain f (dom(f) ⊆ ℝⁿ) is open and f is differentiable at every point of its domain (∀x ∈ dom(f)).