Continuity of additive functions

I was checking this link Proving that an additive function $f$ is continuous if it is continuous at a single point and both Jspecter and Alex Becker's solutions seem to rely on the fact that

$$\lim_{x\rightarrow c} f(x) = \lim_{x\rightarrow a} f(x-a+c)$$
Could someone please explain to me how is that you can change the scalar that x is approaching and still hold the equality? And how can you tell that $$f(x-a+c)$$ is defined?

I'm sorry I didn't ask this question in the original post but I don't have the right to comment right now.

Thank you!

Answer

This is just substituting the variables - instead of $x\to c$ take $x=y-a+c$, or $y=x+a-c$. Then $y\to a$, and we get that

$$\lim_{x\to c}f\left(x\right)=\lim_{x\to c}f\left(x+a-c-a+c\right)=\lim_{y\to a}f\left(y-a+c\right)$$
But instead of renaming the variable as $y$, they kept denoting it $x$.