Postgres 16

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CREATE TABLE customer (
  id SERIAL NOT NULL PRIMARY KEY,
  name TEXT NOT NULL ,
  email TEXT NOT NULL DEFAULT ''
);

CREATE TABLE

INSERT INTO customer(name, email)
VALUES 
  ('Vasya', 'vasya@email.com'), 
  ('Petya', 'petya@email.com');

INSERT 0 2

CREATE OR REPLACE FUNCTION find_customer_name_by_email(email TEXT) 
RETURNS TABLE(name TEXT)
LANGUAGE plpgsql    
AS $$
BEGIN
    RETURN QUERY
    SELECT c.name 
    FROM customer AS c
    WHERE c.email = COALESCE(find_customer_name_by_email.email, c.email);
END;$$

CREATE FUNCTION

SELECT find_customer_name_by_email('petya@email.com');

find_customer_name_by_email

Petya

SELECT 1

SELECT find_customer_name_by_email(null);

find_customer_name_by_email

Vasya

Petya

SELECT 2