Keep walking.
I always learned it as continuer à, but according to Le Figaro, Continuer de is also possible and the difference is:
“Continuee à” is used for an action that begins and continues over time. “Continuer de”, on the other hand, is used to refer to a habit, something you ‘never stop doing’.
What a language!
Hi @paris2020,
Well, what do you know! 
I was not aware of the difference in meaning.
I use both and thinking about it, I guess which one I use depends on phonetics and on context (i.e I’ll use whichever one “sounds” better, depending on the surrounding words) …
I don’t have any data to back this up, but I think that “continuer à” is more common than “continuer de”.
Regarding this specific example:
“Continue de marcher.”, doesn’t bother me, but if I was talking to somebody, I would use “à”, mostly because the word “continue” is imperative 2nd person singular, which you would typically use in less formal situations or with people you know.
(If we go by what Le Figaro is saying, then I think “à” should also be preferred.)
I might use “de”, but if I was speaking, I would definitively shorten it:
“continue d’ marcher”
Now, if a stranger was asking me for directions, I could say something like:
“Tournez à droite, puis continuez (de marcher) jusqu’au feu…”
Again, I would not be thinking about what Le Figaro is saying… In this case, it would be the fact that I am speaking to a stranger. More formal context => more formal register => “continuez + de” more likely.
Fun, is it not?
If “continuer de” sounds kind of weird to you (it sometimes does to me), then maybe this will help:
Take the “weird sounding” sentence and think of what you’d say if you wanted the action to stop instead.
Here are a few examples:
Stop crying! → Arrête de pleurer!
I hope it stops raining. → J’ espère qu’il va s’arrêter de pleuvoir.
He stopped/quit smoking. → Il a arrêté de fumer.