Can a switch hitter change sides during a time at bat?

A batter can change sides of the plate as many times as they choose. Just like a manager can change pitchers as many times as they choose.

No a batter cannot change sides, unless the opposing team brings in a new pitcher that throws with the other hand.

Although managers may make unlimited batter substitutions, the same does not apply to pitchers. Unless he is injured, once a pitcher is brought into a game, he must face at least one batter before he can be taken out. The hitter can switch sides if there is a pitching change, but he can only do it once.

Where is the rule in MLB that states a batter cannot switch sides multiple times when batting? I have seen this answer countless times and no one specifies a rule.

UPDATED INFORMATION:

There is no official rule governing this issue. All the rule books state is that a batter cannot switch sides at the plate when the pitcher is set to deliver a pitch. This means that he/she may switch as often as he/she would like as long as the pitcher is not in his/her motion. There is nothing stating the number of times that he/she may switch sides at a single at bat. That is only a myth.