Consider the following decision: If x < 0.5 go left at a constant speed till the hay is reached, else go right at a constant speed until the hay is reached.

Then A(t,x) is clearly not continuous in x, and we can easily bound the time required to make a decision. The nuance here is that we have to somehow be able to distinguish 0.5 - eps from 0.5 for very small epsilon.

Edit: on further thought, suppose we had a device which measured reasonably well. More precisely it tells us x < 0.5 if x is actually <= 0.5 - c, it tells us x >= 0.5 if x > 0.5 + c, and tells us it is unsure otherwise. We do not know c, but it is deterministic (and hopefully reasonably small). Then we can decide to go left if it tells us x < 0.5, and right if it tells us unsure or that x >= 0.5.