If you’re going through hell, keep on going, Winston Churchill

image info

In one variable, limits and continuity follow the familiar ε–δ pattern. In ℂ we use disks (or balls) instead of intervals, but the ideas carry over verbatim. Then, in ℝⁿ→ℝᵐ, “differentiability” becomes the existence of a unique linear map (the Jacobian) giving the best first‐order approximation. This article builds that bridge between different levels of abstraction in Calculus step by step.

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 continuous at every point in its domain (∀z₀ ∈ D).

Differentiability at a point

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 $\lim_{\substack{z \in \text{dom} f \\ z \neq x, z \to x}} \frac{||f(z) - f(x) - Df(x)(z-x)||_2}{||(z-x)||_2} = 0$ [*]

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

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)).

image info

Figure. For f(x,y)=x²+y², the red plane at (1,1) is the Jacobian’s linear approximation.

Continuous differentiability

Definition. Let $\mathbb{f}:\mathbb{R}^n \to \mathbb{R}$ be a real valued function with domain S = dom(f). Let $\mathbb{U} \subseteq \mathbb{S}$ be an open set. If all the partial derivatives of f exist and are continuous at every point x ∈ U, then f is said to be continuously differentiable on U. If the domain S itself is an open set and f is continuously differentiable on S, then f is said to be continuously differentiable.

A function is continuously differentiable if its derivative (in single-variable calculus) or all its partial derivatives (in multivariable calculus) exist and are themselves continuous functions. This means the function’s rate of change varies smoothly, without abrupt jumps. This concept is crucial because it ensures predictable behavior (they can’t suddenly change slope; it rules out “corners” and makes graphs visually and analytically well-behaved), enables key theorems, etc., e.g, x³ –5x + 2 is C∞ —its derivative 3x²–5 is continuous everywhere; g(x, y) = xy·eˣ. Partials: ∂g/∂x = yeˣ + xy eˣ, ∂g/∂y = x·eˣ. Both are continuous across ℝ², so g is C¹.

1️⃣ In single-variable calculus, a function f: ℝ→ℝ is defined as continuously differentiable (f ∈ C¹(a, b)) on an open interval (a,b) if its derivative f′(x) exists for every point x∈(a,b) and this derivative function f′:(a,b)→R is itself a continuous function. Geometrically, it means that not only does the function have a well-defined tangent at every point within the interval, but the slope of this tangent changes smoothly as you move along the curve, without any abrupt jumps or breaks.

Many common elementary functions are continuously differentiable (C:¹) on their entire domains: polynomial functions (x², 8x⁴ + 3x² -7x + 2), trigonometrical and inverse trigonometrical functions (sin(x) and cos(x) on ℝ, tan(x) on ℝ \ {π/2 + kπ}, arcsin(x) and arccos(x) on (−1, 1), arctan(x) on ℝ), power and logarithmic functions (e^x for all real x, a^x for any base a > 0, ln(x) on (0, ∞), $log_a(x)=\frac{ln(x)}{ln(a)}, \forall x \gt 0$), hyperbolic functions (sinh(x), cosh(x), tanh(x) on ℝ) etc.

The requirement that the interval (a, b) is open ensures that we are dealing with points where the function is well defined in a neighborhood, allowing for the proper definition of limits and continuity of the derivative.

The derivative of a function, f′(x), represents its instantaneous rate of change. If f′(x) is itself a continuous function, it means that this rate of change does not jump suddenly from one value to another, but varies continuously. Instead, small changes in x lead to small changes in f′(x).

This is a stronger condition than mere differentiability. A function can be differentiable at a point, meaning it has a defined rate of change there, but this rate of change might not be a continuous function, e.g.,

$f(x) = \begin{cases} x^2·sin(1/x), &x \ne 0 \\\\ 0, &x = 0 \end{cases}$

f is differentiable everywhere, including at x = 0. f’(0) = $\lim_{h \to 0} \frac{h^2·sin(1/h)-0}{h} = \lim_{h \to 0} h·sin(1/h) = 0$. Note: For every real h ≠ 0, we know $-1 \le \sin \bigl( \frac{1}{h} \bigr) \le 1.$ ∀h > 0, $-h \le \sin \bigl( \frac{1}{h} \bigr) \le h,$ the Squeeze Theorem guarantees that $\lim_{h \to 0+} h·sin(1/h) = 0$. Similarly, ∀h < 0 $-h \ge \sin \bigl( \frac{1}{h} \bigr) \ge h,$ the Squeeze Theorem guarantees that $\lim_{h \to 0-} h·sin(1/h) = 0$.

However, its derivative f′(x) = 2xsin(1/x) −cos(1/x) (for x ≠ 0) does not approach a limit as x→0, so f ′(x) is not continuous at x = 0. This means that while the function f has a tangent at every point, the slope of this tangent oscillates quite wildly near zero, preventing the derivative from being continuous. A C¹ function avoids such erratic behavior in its rate of change.

2️⃣ This concept extends naturally to functions of several variables. Let f : ℝⁿ → ℝ be a real-valued function with domain S = dom(f), and let U ⊆ S be an open set. The function f is said to be continuously differentiable on U if all of its first-order partial derivatives $\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \cdots, \frac{\partial f}{\partial x_n}$ exist and are continuous at every point x ∈ U. If the domain S itself is an open set and f is continuously differentiable on S, then f is simply said to be continuously differentiable.

Specifying $U \subseteq S$ as an open set is crucial, as differentiability is typically defined in open sets where points have neighborhoods entirely contained in the domain, allowing partial derivatives to be evaluated in all directions.

Functions where each variable appears in a polynomial term or a well-behaved trigonometric or exponential term are often continuously differentiable. Consider the function f(x, y) = sin(xy) + e^{x + y}. Its partial derivatives are $\frac{\partial f}{\partial x} = y·cos(xy) + e^{x+y}, \frac{\partial f}{\partial y} = x·cos(xy) + e^{x+y}$. Both of these partial derivatives are continuous functions because they are composed of products, compositions, and sums of continuous functions (polynomial, trigonometric, and exponential functions). Thus, g(x,y) is continuously differentiable on ℝ². These functions represent smooth surfaces without sharp creases or edges, and their tangent planes change orientation continuously as the point of tangency moves.

This definition is really key in multivariable calculus since the existence of partial derivatives alone is not sufficient to guarantee that a function is differentiable in the full sense. However, if the partial derivatives exist and are continuous:

It is a sufficient condition for (total) differentiability. Continuous partial derivatives ensure the function is differentiable in the full sense (i.e., has a linear approximation via the Jacobian).

A function f : ℝⁿ → ℝ is differentiable at a point x₀ ∈ U if there exists a linear map (represented by the Jacobian matrix Df(x₀)) such that: f(x₀ + h) = f(x₀) + Df(x₀)·h + o(||h||) where o(||h||) is the remainder term that vanishes faster than o||h|| as h → 0.

Continuity of partial derivatives ensures the function’s behavior is smooth enough for the linear approximation to hold.

Example: f(x, y) = x² + y². Partial derivatives: $\frac{\partial f}{\partial x} = 2x, \frac{\partial f}{\partial y} = 2y$. Both 2x and 2y are continuous everywhere (as they are linear functions), so f is continuously differentiable on $\mathbb{R}^2$, an open set. The Jacobian is $\mathbb{Df}(1, 1) = [2, 2]$, and the function is differentiable, with the linear approximation: $f(1+h, 1+k) \approx 2 + 2h + 2k.$

Continuous differentiability ensures a well-defined tangent plane. In higher dimension (ℝⁿ), the tangent plane becomes a tangent hyperplane, defined similarly using the Jacobian.

In ℝ², for a function f(x, y), the tangent plane at a point $(x_0, y_0)$ is: $z = f(x_0, y_0) + \frac{\partial f}{\partial x}(x_0, y_0)(x - x_0) + \frac{\partial f}{\partial y}(x_0, y_0)(y - y_0).$ In our example, f(1, 1) = 1² + 1² = 2, and the partial derivatives are $\frac{\partial f}{\partial x}(x_0, y_0) = 2·x(1, 1) = 2, \frac{\partial f}{\partial y}(x_0, y_0) = 2$. The tangent plane is: z = 2 + 2(x - 1) + 2(y - 1) = 2x + 2y - 2.

It ensures a smooth (consistent) behavior of the function’s rate of change in all coordinate directions simultaneously.

The partial derivatives $\frac{\partial f}{\partial x_i}$ represent the rate of change of f in each coordinate direction. Continuity of these derivatives ensures that the rate of change varies smoothly across U, avoiding abrupt jumps or discontinuities.

Example: f(x, y) = x² + y². Partial derivatives: $\frac{\partial f}{\partial x} = 2x, \frac{\partial f}{\partial y} = 2y$. Both 2x and 2y are continuous everywhere, so the rate of change (slope) in the x- and y- directions changes smoothly. At (1, 1) the slopes are both 2, and nearby (e.g., at (1.1, 1.1)) they are 2·1.1 = 2.2 a small, smooth change.

The function is C¹, the class of functions whose first-order partial derivatives exist and are continuous on their domain, e.g., f(x, y) = x² + y² is C¹ on ℝ².

This property is significant because it implies differentiability and smoothness, enabling results like the existence of tangent planes and linear approximations.