Learn Finnish: Tie jatkuu rajalle asti. - The road continues as far as the border.

The postposition asti (as far as / up to) requires the word before it to be in a directional case, like the illative ("into") or allative ("onto/to"). Because you usually travel to a border (rajalle), it stays in the allative case when you add asti to mean "all the way to the border".

Could I use saakka instead of asti in this sentence?

Yes, Tie jatkuu rajalle saakka is perfectly correct. The postpositions asti and saakka are completely interchangeable when talking about physical limits or points in time.

Why use the verb jatkuu instead of jatkaa for "continues"?

Jatkua (jatkuu) is an intransitive verb, meaning something continues on its own state (like a road, a movie, or the weather). Jatkaa is transitive, used when a person actively continues an action or an object, like continuing a journey. Since the road isn't "doing" an action to an object, we use jatkuu.

Would it be wrong to say rajaan asti using the illative case instead?