Can you negate continuous spell cards?

You can negate the continuous spell card's initial activation with cards like Solemn Judgment or Magic Jammer.

Once successfully activated though, you can't use 'spell card activation' negation effects like the above, even if the continuous spell has a subsequent activated effect, like the trigger of Black Whirlwind, or Royal Oppression's Quick Effect. That's because the spell card's activation was when you first played it to the field, what you are doing after is 'activating an effect' of an already resolved card. However effect negation can be chained to it, you could use Stardust Dragon against Royal Oppression (as long as Royal Oppression is destroying a card on the field - usually it does not, but it would if it was being used against Summon Priest).

Lastly, cards like Spell Canceller, and Imperial Order (forbidden in Advanced format) will 'switch off' any continuous cards, and negate any of their activated effects too.