The Callback Arguments section of the f2py docs here contains a non-working example.

If we follow the docs and compile the callback.f file using f2py -c -m callback callback.f, the following happens when we try to reproduce the python session:

>>> import callback
>>> def f(i): return i*i
... 
>>> print(callback.foo(f))
0.0
>>> 

The expected result was 110. Changing the Fortran file to declare FUN as a REAL*8 argument does the trick (but I don't understand exactly why this is needed and it's not mentioned in the docs)

C FILE: CALLBACK.F
      SUBROUTINE FOO(FUN,R)
      EXTERNAL FUN
      INTEGER I
      REAL*8 R
      REAL*8 FUN
Cf2py intent(out) r
      R = 0D0
      DO I=-5,5
         R = R + FUN(I)
      ENDDO
      END
C END OF FILE CALLBACK.F

Compile again using f2py -c -m callback callback.f and we get

>>> import callback
>>> def f(i): return i*i
... 
>>> print(callback.foo(f))   
110.0

I'm guessing an explanation of this should be added in the docs. I'm working on updating them but I need help to explain this behaviour. Any input is appreciated!

>>> import sys, numpy; print(numpy.__version__, sys.version)
1.17.3 3.6.7 | packaged by conda-forge | (default, Jul  2 2019, 02:18:42) 
[GCC 7.3.0]
>>>