Animated Solution for Mathematics - Limits, Continuity and Differentiability: Let $f(x+y)=f(x)+f(y)$ for all $x$ and $y$. If the function $f(x)$ is continuous at $x=0$, then show that $f(x)$ is continuous at all $x$.

Welcome, fellow traveler, to one of the most elegant corners of mathematics. Today, we are going to unravel a classic: Cauchy's Functional Equation.

You have likely seen the equation $f(x+y)=f(x)+f(y)$ before, but have you ever stopped to appreciate what it is actually telling you? It is a promise; a rule that says the function respects the structure of addition.

If you add two inputs, the outputs add up as well. This is the hallmark of linearity. But how do we prove that this simple rule forces the function to be continuous everywhere, provided it is continuous at just one point? Let us embark on this journey.

The Anchor at the Origin

Before we can talk about continuity at any point $x$, we must understand the function's behavior at the very beginning—the origin. Let us test the equation with the simplest possible inputs.

If we set $x=0$ and $y=0$, the equation $f(x+y)=f(x)+f(y)$ becomes:

This simplifies to $f(0)=2f(0)$, which immediately forces $f(0)=0$. Think of this as our anchor. We now know for a fact that our function must pass through the origin $(0,0)$. This is the first piece of the puzzle, and it is non-negotiable.

The Definition of Continuity

Now, let us look at the definition of continuity. To say a function $f(x)$ is continuous at a point $x$, we mean that as we approach $x$ from any direction, the value of the function approaches $f(x)$.

Mathematically, we write this as:

The problem gives us a gift: we know $f(x)$ is continuous at $x=0$. By definition, this means ${\lim }_{h\to 0}f(0+h)=f(0)$. Since we just proved $f(0)=0$, this simplifies to ${\lim }_{h\to 0}f(h)=0$. This is our 'local' knowledge; we know exactly what happens when we are infinitesimally close to the origin.

The Bridge to Global Continuity

Here is where the magic happens. We want to prove continuity at any arbitrary point $x$. We start with the limit definition: ${\lim }_{h\to 0}f(x+h)$.

We cannot evaluate this directly, but we have our functional equation! We can rewrite $f(x+h)$ as $f(x)+f(h)$. Now, our limit becomes:

Because $f(x)$ does not depend on $h$, we can pull it out of the limit: $f(x)+{\lim }_{h\to 0}f(h)$.

The Final Cancellation

Look at what we have achieved. We have transformed the limit of the function at $x+h$ into a sum involving the limit of the function at $h$.

We already know from Phase 2 that ${\lim }_{h\to 0}f(h)=0$. Substituting this in, we get $f(x)+0$, which is simply $f(x)$.

Thus, we have shown that ${\lim }_{h\to 0}f(x+h)=f(x)$. By definition, this means the function is continuous at every point $x$. We have successfully bridged the gap from local continuity at the origin to global continuity everywhere. It is a beautiful demonstration of how a single constraint can dictate the behavior of an entire system.