How to find if a function exists in PostgreSQL?

I think the easiest way would be to use pg_get_functiondef().

If it returns something, the function is there, otherwise the function does not exist:

select pg_get_functiondef('some_function()'::regprocedure);
select pg_get_functiondef('some_function(integer)'::regprocedure);

The drawback is that it will produce an error if the function isn't there instead of simply returning an empty result. But this could e.g. be overcome by writing a PL/pgSQL function that catches the exception and returns false instead.

Yes, you cannot to find functions in pg_class because functions are stored on system table pg_proc

postgres-# \df
                               List of functions
 Schema |        Name        | Result data type | Argument data types  |  Type  
--------+--------------------+------------------+----------------------+--------
 public | foo                | integer          | a integer, b integer | normal
 public | function_arguments | text             | oid                  | normal
(2 rows)

Query for list of custom functions based on pg_proc is simply

postgres=# select p.oid::regprocedure
              from pg_proc p 
                   join pg_namespace n 
                   on p.pronamespace = n.oid 
             where n.nspname not in ('pg_catalog', 'information_schema');
           oid           
-------------------------
 foo(integer,integer)
 function_arguments(oid)
(2 rows)

Most simply and fastest tests on functions existence are casting (without parameters) to regproc or regprocedure (with parameters):

postgres=# select 'foo'::regproc;
 regproc 
---------
 foo
(1 row)

postgres=# select 'foox'::regproc;
ERROR:  function "foox" does not exist
LINE 1: select 'foox'::regproc;
               ^
postgres=# select 'foo(int, int)'::regprocedure;
     regprocedure     
----------------------
 foo(integer,integer)
(1 row)

postgres=# select 'foo(int, text)'::regprocedure;
ERROR:  function "foo(int, text)" does not exist
LINE 1: select 'foo(int, text)'::regprocedure;
               ^

or you can do some similar with test against pg_proc

postgres=# select exists(select * from pg_proc where proname = 'foo');
 exists 
--------
 t
(1 row)

postgres=# select exists(select * 
                            from pg_proc 
                           where proname = 'foo' 
                             and function_arguments(oid) = 'integer, integer');
 exists 
--------
 t
(1 row)

where:

CREATE OR REPLACE FUNCTION public.function_arguments(oid)
RETURNS text LANGUAGE sql AS $function$
    select string_agg(par, ', ') 
       from (select format_type(unnest(proargtypes), null) par 
                from pg_proc where oid = $1) x
$function$

or you can use buildin functions:pg_get_function_arguments

p.s. trick for simply orientation in system catalog. Use a psql option -E:

[pavel@localhost ~]$ psql -E postgres
psql (9.2.8, server 9.5devel)
Type "help" for help.

postgres=# \df
********* QUERY **********
SELECT n.nspname as "Schema",
  p.proname as "Name",
  pg_catalog.pg_get_function_result(p.oid) as "Result data type",
  pg_catalog.pg_get_function_arguments(p.oid) as "Argument data types",
 CASE
  WHEN p.proisagg THEN 'agg'
  WHEN p.proiswindow THEN 'window'
  WHEN p.prorettype = 'pg_catalog.trigger'::pg_catalog.regtype THEN 'trigger'
  ELSE 'normal'
END as "Type"
FROM pg_catalog.pg_proc p
     LEFT JOIN pg_catalog.pg_namespace n ON n.oid = p.pronamespace
WHERE pg_catalog.pg_function_is_visible(p.oid)
      AND n.nspname <> 'pg_catalog'
      AND n.nspname <> 'information_schema'
ORDER BY 1, 2, 4;
**************************

                               List of functions
 Schema |        Name        | Result data type | Argument data types  |  Type  
--------+--------------------+------------------+----------------------+--------
 public | foo                | integer          | a integer, b integer | normal
 public | function_arguments | text             | oid                  | normal
(2 rows)