Question

In *.h header files of a C library, should one declare functions

extern void f();

// or only 

void f();

1. when using only in C
2. when using from C++.

Answer 1

There’s [almost] never any need to use the keyword extern when declaring a function, either in C or in C++. In C and in C++ all functions have external linkage by default. The strange habit of declaring functions in header files with extern probably has some historical roots, but it has been completely irrelevant for decades already.

There’s one [obscure?] exception from the above in C, which is probably not directly related to what you are asking about: in C language (C99) if in some translation unit a function is defined as inline and also declared as extern (an explicit extern is used) then the inline definition of that function also serves as an external definition. If no declarations with explicit extern are present in the translation unit, then the inline definition is used as “internal” definition only.

P.S. There’s such thing as extern "C" in C++, but that is a completely different matter.