Would you call with A high on the river facing villain's half pot bet after you checked? More details: Villain just called your bet from pre-flop to turn.
With this information only, I wouldn't call. Did I understand it correctly that you raised on all streets that far? If so, they got you if you only have A high. They only need one pair, no matter how small, to beat you in that situation. Or even just a higher kicker unless yours was King. So that's my basic answer without any further information. For a better analysis we would need to know at least your kicker and the board texture. Preferably positions and playing styles as well. However, even with that information I would hardly never call with an A high if my opponent has been calling my raises from preflop to river.
EDIT: The only time I would consider continuing on river is if I would have a spot-on read that my opponent was drawing a flush and didn't got it on river. But even then I wouldn't just call but instead I would continue to
bluff and reraise big. Still, quite risky.