Cauchy's functional equation

Cauchy's Functional Equation is a math problem where we have a function that we need to figure out. The problem says that if we have two numbers, let's call them "a" and "b", then when we add them together, the output of the function is the same as if we added the outputs of the function when we use "a" and "b" separately.

Let's pretend that our function is like a pizza maker. When we give the pizza maker two different toppings (let's call them "pepperoni" and "mushrooms"), we get one pizza. But if we give the pizza maker the "pepperoni" topping and the "mushroom" topping separately, then we get two smaller pizzas that, when we add them together, equal the one big pizza. So the pizza maker follows Cauchy's Functional Equation!

But there's a catch. The function can be any type of function, not just a pizza maker. And it can be very tricky to figure out what the function actually is. Mathematicians have studied Cauchy's Functional Equation for a long time and have found that there are many different functions that follow this rule, and they all look pretty different from each other.

So, to sum it up: Cauchy's Functional Equation is a math problem where we have to find a function that gives the same output when we add two numbers together as if we added the outputs of the function when we use those two numbers separately. It's like a pizza maker who makes one pizza with two toppings or two smaller pizzas with each topping.