> this is an error because the compiler treats each .c file separately, so when it’s compiling main.c, it isn’t looking at f.c, so it has no way to know the fun

> this is an error because the compiler treats each .c file separately, so when it’s compiling main.c, it isn’t looking at f.c, so it has no way to know the function f exists. But it's not an error and the compiler does not need to know the function exists. If there's no declaration, then the function's type is implicit and simply assumed to exist. If the function doesn't actually exist, then it's your linker that will complain.