As several people have said, the original (with a comma) is perfectly grammatical. However, when you switch it around, it sounds better with "on."

To the point that I became dizzy, I kept on studying.

but I'd be more likely to reword the switched-around version with an "until"

Until I became dizzy, I kept on studying.

or, maybe keeping closer to the meaning of the original,

Until it made me dizzy, I kept on studying.

As several people have said, the original (with a comma) is perfectly grammatical. However, when you switch it around, it sounds better with "kept on".

To the point that I became dizzy, I kept on studying.

and I'd be more likely to reword the switched-around version with an "until"

Until I became dizzy, I kept on studying.

or, maybe keeping closer to the meaning of the original,

Until it made me dizzy, I kept on studying.