gdb disassemble/rs to show source and raw bytes as well
With this format, it gets really close to objdump -S output:
gdb -batch -ex "disassemble/rs $FUNCTION" "$EXECUTABLE"
main.c
#include <assert.h>
int myfunc(int i) {
    i = i + 2;
    i = i * 2;
    return i;
}
int main(void) {
    assert(myfunc(1) == 6);
    assert(myfunc(2) == 8);
    return 0;
}
Compile and disassemble
gcc -O0 -ggdb3 -std=c99 -Wall -Wextra -pedantic -o main.out main.c
gdb -batch -ex "disassemble/rs myfunc" main.out
Disassembly:
Dump of assembler code for function myfunc:
main.c:
3       int myfunc(int i) {
   0x0000000000001135 <+0>:     55      push   %rbp
   0x0000000000001136 <+1>:     48 89 e5        mov    %rsp,%rbp
   0x0000000000001139 <+4>:     89 7d fc        mov    %edi,-0x4(%rbp)
4           i = i + 2;
   0x000000000000113c <+7>:     83 45 fc 02     addl   $0x2,-0x4(%rbp)
5           i = i * 2;
   0x0000000000001140 <+11>:    d1 65 fc        shll   -0x4(%rbp)
6           return i;
   0x0000000000001143 <+14>:    8b 45 fc        mov    -0x4(%rbp),%eax
7       }
   0x0000000000001146 <+17>:    5d      pop    %rbp
   0x0000000000001147 <+18>:    c3      retq   
End of assembler dump.
Tested on Ubuntu 16.04, GDB 7.11.1.
objdump + awk workarounds
Print the paragraph as mentioned at: https://unix.stackexchange.com/questions/82944/how-to-grep-for-text-in-a-file-and-display-the-paragraph-that-has-the-text
objdump -d main.out | awk -v RS= '/^[[:xdigit:]]+ <FUNCTION>/'
e.g.:
objdump -d main.out | awk -v RS= '/^[[:xdigit:]]+ <myfunc>/'
gives just:
0000000000001135 <myfunc>:
    1135:   55                      push   %rbp
    1136:   48 89 e5                mov    %rsp,%rbp
    1139:   89 7d fc                mov    %edi,-0x4(%rbp)
    113c:   83 45 fc 02             addl   $0x2,-0x4(%rbp)
    1140:   d1 65 fc                shll   -0x4(%rbp)
    1143:   8b 45 fc                mov    -0x4(%rbp),%eax
    1146:   5d                      pop    %rbp
    1147:   c3                      retq   
When using -S, I don't think there is a fail-proof way, as the code comments could contain any possible sequence... But the following works almost all the time:
objdump -S main.out | awk '/^[[:xdigit:]]+ <FUNCTION>:$/{flag=1;next}/^[[:xdigit:]]+ <.*>:$/{flag=0}flag'
adapted from: How to select lines between two marker patterns which may occur multiple times with awk/sed
Mailing list replies
There is a 2010 thread on the mailing list which says it is not possible: https://sourceware.org/ml/binutils/2010-04/msg00445.html
Besides the gdb workaround proposed by Tom, they also comment on another (worse) workaround of compiling with -ffunction-section which puts one function per section and then dumping the section.
Nicolas Clifton gave it a WONTFIX https://sourceware.org/ml/binutils/2015-07/msg00004.html , likely because the GDB workaround covers that use case.