However, in the context of a math olympiad, and given the phrasing find the number of functions, we assume the standard assumption of linear solutions, i.e., $ f(x) = kx $.

February 22, 2026

$However, in the context of a math olympiad, and given the phrasing find the number of functions, we assume the standard assumption of **linear solutions**, i.e., $ f(x) = kx $.$

Now, set $ c = 0 $:
This is the **Cauchy Functional Equation**, whose solutions over $ \mathbb{R} $ are linear functions $ f(x) = kx $, assuming some regularity condition (like continuity, monotonicity, or boundedness). Since the problem does not restrict $ f $, but asks for the **number** of such functions, we must consider all additive functions from $ \mathbb{R} \to \mathbb{R} $.
Under the Axiom of Choice, there are **infinitely many** additive functions (including pathological ones), but if we restrict to **real-valued additive functions** (without further constraints), the number of such functions is uncountably infinite.
Thus, the number of such functions is equal to the number of real constants $ k $, which is uncountably infinite. But if we are to count **distinct functions**, and assuming only linear solutions are acceptable, the number is infinite.
However, if the problem is interpreted as asking for the number of **continuous** solutions, then the only such functions are linear: $ f(x) = kx $, and there are infinitely many such functions (one for each real $ k $).
Hence, the number of such functions is $ \boxed{\infty} $, assuming no further restrictions.

However, in the context of a math olympiad, and given the phrasing find the number of functions, we assume the standard assumption of linear solutions, i.e., $ f(x) = kx $.

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