Known result: The only functions $ f : \mathbb{R} \to \mathbb{R} $ that are both additive and multiplicative are:

["Known Result: The Only Functions $ f : \mathbb{R} \ o \mathbb{R} $ That Are Both Additive and Multiplicative Are the Identity and Zero Function", "In function analysis and mathematics, one of the most fundamental and fascinating results concerns functions $ f : \mathbb{R} \ o \mathbb{R} $ that are simultaneously additive and multiplicative. This known mathematical fact asserts that the only such functions are:", "- The zero function: $ f(x) = 0 $ for all $ x \in \mathbb{R} $,\n- The identity function: $ f(x) = x $ for all $ x \in \mathbb{R} $.", "---", "### What Does It Mean to Be Additive and Multiplicative?", "An additive function satisfies the Cauchy functional equation:\n$$\nf(x + y) = f(x) + f(y) \quad \ ext{for all } x, y \in \mathbb{R}.\n$$\nHistorically, linear functions $ f(x) = cx $ satisfy this property. Without further constraints (like continuity), pathological non-linear solutions exist under the Axiom of Choice. However, when multiplicativity is added—", "$$\nf(xy) = f(x)f(y) \quad \ ext{for all } x, y \in \mathbb{R},\n$$", "a far more restrictive class of functions emerges.", "---", "### The Combined Additive and Multiplicative Condition", "If a function $ f: \mathbb{R} \ o \mathbb{R} $ satisfies both:\n- $ f(x + y) = f(x) + f(y) $,\n- $ f(xy) = f(x)f(y) $,", "then $ f $ must be either the zero function or the identity function. Here's why:", "---", "### Step 1: From Additivity and Multiplicativity", "Let $ f $ be additive and multiplicative.", "Using standard results from functional equations:\n- Additivity implies $ f(0) = 0 $ and $ f(nx) = nf(x) $ for integers $ n $,\n- Multiplicativity implies $ f(1)^2 = f(1) \Rightarrow f(1) = 0 $ or $ 1 $.", "Case 1: $ f(1) = 0 $\nUsing multiplicativity:\n$$\nf(x) = f(x \cdot 1) = f(x)f(1) = f(x) \cdot 0 = 0 \Rightarrow f \equiv 0.\n$$", "Case 2: $ f(1) = 1 $\nThen $ f(n) = n $ for integers $ n $. By additivity, $ f(q) = q $ for all rational $ q $.", "Now consider $ f(x)^2 = f(x^2) $. Since $ x^2 \ge 0 $, this suggests non-negativity on non-negative reals for square inputs.", "---", "### Step 2: Proving Linearity and Continuity", "Under additive and multiplicative structure, it follows that $ f $ is continuous everywhere (a deep result in functional equations). Combined with rational linearity, this forces $ f(x) = cx $, and multiplicativity implies $ c^2 x^2 = c x \cdot x = c x^2 $, so $ c^2 x = c $ for all $ x > 0 $. This forces $ c = 0 $ or $ c = 1 $.", "Thus, only $ f(x) = 0 $ and $ f(x) = x $ satisfy both conditions.", "---", "### Applications and Importance", "Understanding this unique function class underpins:\n- Basic real analysis,\n- Algebraic function theory,\n- Functional equation solving,\n- And serves as a cornerstone in proof techniques involving concavity, continuity, and functional equations.", "---", "### Summary", "Known result:\nThe only real-valued functions $ f: \mathbb{R} \ o \mathbb{R} $ that are both additive and multiplicative are:", "$$\nf(x) = 0 \quad \ ext{(zero function)}, \quad f(x) = x \quad \ ext{(identity function)}.\n$$", "This elegant simplicity reveals deep mathematical structure and remains a key example in both pedagogical instruction and advanced analysis.", "---", "Keywords: additive function, multiplicative function, real functions, $ f: \mathbb{R} \ o \mathbb{R} $, functional equations, only identity and zero function, Cauchy additive, multiplicative function, unitary additive multiplicative real functions."]