What is the 'function' keyword used in some bash scripts?

For example: Bash-Prog-Intro-HOWTO

function foo() {}

I make search queries in info bash and look in releted chapters of POSIX for function keyword but nothing found.

What is function keyword used in some bash scripts? Is that some deprecated syntax?

The function keyword is optional when defining a function in Bash, as documented in the manual:

> Functions are declared using this syntax: > > name () compound-command [ redirections ] > > or > > function name [()] compound-command [ redirections ]

The first form of the syntax is generally preferred because it's compatible with Bourne/Korn/POSIX scripts and so more portable.
That said, sometimes you might want to use the function keyword to prevent Bash aliases from colliding with your function's name. Consider this example:

$ alias foo="echo hi"
$ foo() { :; }
bash: syntax error near unexpected token `('

Here, 'foo' is replaced by the text of the alias of the same name because it's the first word of the command. With function the alias is not expanded:

 $ function foo() { :; }

The function keyword is necessary in rare cases when the function name is also an alias. Without it, Bash expands the alias before parsing the function definition -- probably not what you want:

alias mycd=cd
mycd() { cd; ls; }  # Alias expansion turns this into cd() { cd; ls; }
mycd                # Fails. bash: mycd: command not found
cd                  # Uh oh, infinite recursion.

With the function keyword, things work as intended:

alias mycd=cd
function mycd() { cd; ls; }  # Defines a function named mycd, as expected.
cd                           # OK, goes to $HOME.
mycd                         # OK, goes to $HOME.
\mycd                        # OK, goes to $HOME, lists directory contents.

The reserved word function is optional. See the section 'Shell Function Definitions' in the bash man page.