Continuity of a current through an inductor
so where is this &quot;infinite current&quot; coming from?

Well, I'm sad to report that it appears to be coming from your imagination. This is because the website you linked doesn't relate infinite current with an inductor at all. However, it does say this: -

The principle of continuity of capacitive voltage says: In the absence
of infinite current, the voltage across a capacitor cannot change
instantaneously.
The dual of this is the principle of continuity of inductive current:
In the absence of infinite voltage, the current through an inductor
cannot change instantaneously.

In other words, it talks about infinite current in the context of a capacitor but, it only talks about infinite voltage in the context of an inductor.

After a question amendment by the OP....

so where is this &quot;infinite voltage&quot; coming from?

It's coming from this equation: -
$$V = L\cdot\dfrac{di}{dt}$$
You have created a current discontinuity by opening SW3 and that means \$\frac{di}{dt}\$ becomes infinite and that means the voltage has to become infinite.