UNIQUE Continuous Function PROOF!
We prove that a continuous real-valued function of one real variable that satisfies the property f(x+y) = f(x)f(y) is uniquely determined by the value f(1)! In the process, we explain how to think about rigorous mathematics and introduce mathematical proofs and how mathematicians discover them. The proof could be considered part of the advanced mathematics subject of real analysis, although it can be understood with only a knowledge of what are continuous functions.
If f(1) = a, then we prove that the function is necessarily the exponential function (or power function) f(x) = a^x (the fact that this function satisfies the property follows from exponent laws). The proof is in steps showing how we can construct the values of f(x) first for natural numbers, then integers, then rational numbers using only the property and the value f(1). Finally, we use the continuity of f to construct the values of f(x) for all real numbers. In rigorous mathematical language, we say that the function f is a continuous group homomorphism from the additive group of the real numbers to the multiplicative group of non-zero real numbers, and we prove that such a function uniquely determined by the value f(1).
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