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If I get Pithing Needle onto the battlefield in response (giving it flash or getting it out at instant speed) to an activated ability being activated, and chose the thing that's activated ability is being activated, what happens?

Example:

1. Enemy activates Teferi, Hero of Dominaria's ability.


2. I cast Whir of Invention in response.


3. The stack starts to resolve, and my Whir of Invention grabs me my Pithing Needle.


4. I name Teferi, Hero of Dominaria with my needle.


5. What happens?

The Pithing Needle does nothing.

602.2. To activate an ability is to put it onto the stack and pay its costs, so that it will eventually resolve and have its effect. Only an object’s controller (or its owner, if it doesn’t have a controller) can activate its activated ability unless the object specifically says otherwise. Activating an ability follows the steps listed below, in order. If, at any point during the activation of an ability, a player is unable to comply with any of those steps, the activation is illegal; the game returns to the moment before that ability started to be activated (see rule 721, “Handling Illegal Actions”). Announcements and payments can’t be altered after they’ve been made.

Once an ability is on the stack it has already been activated. Effects that would prevent the player from activating that ability no longer matter. The ability on the stack will resolve as normal.

Pithing Needle only prevents future activations of the chosen permanent's abilities once it's on the battlefield. It has no effect on the chosen permanent for previous activations of its abilities, even if an ability of that permanent is currently on the stack.

Pithing Needle's static ability creates a continuous effect. It is in effect as soon as and as long as the Needle is on the battlefield, but not before.

611.3. A continuous effect may be generated by the static ability of an object.

611.3b The effect applies at all times that the permanent generating it is on the battlefield or the object generating it is in the appropriate zone.

Pithing Needle prevents the activation of a permanent's ability. The act of activating an ability is a process that, once finished, can't be prevented retroactively. Activated abilities can be countered by certain cards, just like spells, but Pithing Needle does not counter abilities.

602. Activating Activated Abilities

602.2. To activate an ability is to put it onto the stack and pay its costs, so that it will eventually resolve and have its effect. [..]

This is the rule specifically describing the Needle's effect. However, by the time your opponent has activated the ability, the ability was not prohibited from being activated.

602.5. A player can’t begin to activate an ability that’s prohibited from being activated.