So $ f $ is additive. Also, from the multiplicative condition:

["Understanding the Fundamental Property: How $ f $ Being Additive and Multiplicative Shapes Mathematical Functions", "In the world of advanced mathematics—especially within algebra, functional equations, and number theory—function behavior often follows precise, elegant rules. One particularly interesting class of functions satisfies both additivity and a multiplicative condition derived from it. This article explores the powerful implications of a function $ f $ being additive, how it connects with multiplicative properties, and why this duality is important across mathematical disciplines.", "---", "### What Does It Mean for $ f $ to Be Additive?", "A function $ f: D \ o \mathbb{R} $ (or another field) is called additive if, for all $ x, y $ in its domain satisfying $ x + y \in D $, the following holds:\n[\nf(x + y) = f(x) + f(y)\n]", "Additivity is a strong linear property that, in many spaces, characterizes linearity—particularly over fields like $ \mathbb{R} $. Without additional constraints (like continuity), additive functions can exhibit pathological behavior (as shown by Cauchy’s functional equation and non-measurable solutions). However, under mild regularity assumptions, additivity implies strong structure.", "For functions defined on $ \mathbb{R} $ with domain closed under addition and continuity (or measurability), we know:\n- $ f(0) = 0 $\n- $ f(nx) = n f(x) $ for all integers $ n $\n- $ f(qx) = q f(x) $ for rational $ q $\n- $ f $ is linear: $ f(x) = kx $ for some constant $ k $", "---", "### The Multiplicative Condition and Functional Consequences", "Beyond additivity, the function $ f $ often satisfies an additional multiplicative condition:\n[\nf(xy) = f(x)f(y)\n]\nfor $ x, y $ in its domain (assuming $ xy $ is defined and remains in the domain).", "This multiplicative property, combined with additivity, imposes striking structural constraints.", "When a function satisfies both\n1. $ f(x+y) = f(x) + f(y) $\n2. $ f(xy) = f(x)f(y) $", "it is said to be both additive and multiplicative—a rare and powerful combination. What does this mean, and which functions satisfy both?", "---", "### Special Functions: The Only Continuous Solutions", "Consider solutions over $ \mathbb{R} $:", "- Solutions to Cauchy’s additive equation $ f(x+y) = f(x) + f(y) $ include linear functions $ f(x) = kx $.\n- The multiplicative condition $ f(xy) = f(x)f(y) $ restricts $ f $ to functions where products behave like scalar multiplication.\n- Only functions of the form $ f(x) = x^k $ (for constant $ k $) satisfy both additivity and multiplicativity over $ \mathbb{R} $ and nonzero rationals — but such power functions only satisfy both equations if $ k = 0 $ or $ k = 1 $.", "Testing $ f(x) = x $:\n[\nf(x+y) = x + y = f(x) + f(y), \quad f(xy) = xy = f(x)f(y)\n]\n✔️ Satisfies both.", "Testing $ f(x) = x^2 $:\n[\nf(x+y) = (x+y)^2 <br/>\ne x^2 + y^2 = f(x) + f(y) \quad \ ext{(fails additivity)}\n]\nThis shows that only the linear identity function satisfies both properties globally over the reals.", "---", "### Importance in Algebra and Number Theory", "Functions that are both additive and multiplicative are rare but deeply connected to fundamental algebraic structures:\n- Such functions are ring homomorphisms from $ \mathbb{R} $ to itself — preserving both addition and multiplication.\n- These maps are crucial in defining natural structures in analysis and algebra, especially in defining norms and valuations.\n- In finite fields or discrete settings, analogous multiplicative-additive functions help define field automorphisms and character sums.", "The only continuous solutions are $ f(x) = x $ and $ f(x) = 0 $. Without continuity, exotic solutions exist (using Hamel bases), but they are non-constructive and non-measurable.", "---", "### Real-World Applications and Mathematical Intuition", "Though abstract, the combination of additivity and multiplicativity appears in:\n- Normalization in probability and statistics: Linear transformations preserving multiplicative structure under constraints.\n- Cryptanalysis and coding theory: Functional equations model signal transformations respecting both linear superposition and multiplicative scaling.\n- Quantum mechanics and linear algebra: Operators preserving vector space operations with multiplicative invariance.", "---", "### Summary: Why This Property Matters", "A function $ f $ that is both additive and multiplicative—under reasonable regularity conditions—must be linear and multiplicative, collapsing to either $ f(x) = x $ or $ f(x) = 0 $. This duality serves as a powerful filtering mechanism in functional analysis, number theory, and applied mathematics.", "Understanding such constraints helps identify invariants, simplify models, and uncover deep symmetries in mathematical systems. Whether studying Fourier analyses, algebraic structures, or dynamical systems, recognizing additive and multiplicative functions unlocks deeper insight.", "---", "### Further Reading", "- Euclid’s Elements — Foundations of multiplicative functions\n- Introduction to Functional Equations by Polya and Szegő\n- Linear Algebra and Its Applications by Gilbert Strang — on linear maps and structure\n- Real and Complex Analysis by Rudin — on functional equations and continuity", "---", "Keywords: additive function, multiplicative function, Cauchy functional equation, linear functions, functional equations, mathematical structure, homomorphism, Cauchy’s identity, real analysis.\nMeta Description: Explore why functions that are both additive and multiplicative must be linear, and uncover the mathematical significance of $ f(x+y) = f(x)+f(y) $ and $ f(xy) = f(x)f(y) $ across algebra and applied sciences.", "---", "Unlock the elegance of mathematical structure—where additivity and multiplicativity converge to define function symmetry and power."]