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).
If you enjoy this video, I would love if you could please like ✅ , subscribe 🎉, and share ⏩ with students, classmates, friends and family - it means so much to help support me in my goal of creating elite infinite free accessible math education worldwide! 😊 If you would like to support my channel beyond this (which of course would mean so much to me 🤗), please check out my "Buy me a Coffee" and my "Patreon" page!
Buy me a Hot Chocolate 😊: https://buymeacoffee.com/math_mastery... ☕️
Become a Patreon Member 🤩: / mathmasterywithamitesh
On this page of the site you can watch the video online UNIQUE Continuous Function PROOF! with a duration of hours minute second in good quality, which was uploaded by the user Math Mastery with Amitesh 07 November 2024, share the link with friends and acquaintances, this video has already been watched 3,528 times on youtube and it was liked by 154 viewers. Enjoy your viewing!
