Why is stddef.h not in /usr/include?

Question

I have compiled the gnu standard library and installed it in $GLIBC_INST.

Now, I try to compile a very simple programm (using only one #include

Answer 1

That's because files under /usr/include are common headers that provided by the C library, for example, glibc, while the files at /usr/lib/gcc are specific for that particular compiler. It is common that each compiler has their own different implementation of stddef.h, but they will use the same stdio.h when links to the installed C library.