[kevhito]

The missing piece of "just choose to always go left" is that this is a degenerate and uninteresting case. No decision is being made.

The range must be discrete and at least 2 possible values.

There is nothing about optimality at all here. Even if both possible outputs are equally "optimal", there is no procedure to pick one in a finite amount of time.

[Kranar]

I doubt the issue has anything to do with being interesting or not. Do you know this for a fact or can give me anything to follow up on what it means for a decision to be interesting? For example if a decision procedure required the choice of chocolate or vanilla depending on the current time of day (a continuous input resulting in a discrete output), and I present an algorithm that categorically chooses chocolate regardless of what time it is, I doubt you'd turn around and claim that my algorithm is not making a decision because it always chooses chocolate.

If an algorithm takes an input, continuous or discrete, discards that input and returns a constant value, that algorithm is just as "legitimate" as any other algorithm in so far as being an algorithm is concerned. It may not produce the optimal output for some given cost, but it is a formal and rigorous decision procedure that manages to output a discrete value given a continuous input in a bounded amount of time.

At any rate, reading the comments when this was last posted it looks like the issue has to do with producing the optimal output in a given time frame. If an optimal discrete decision must be made in the next T seconds on the basis of a continuous input X, then there always exists some value of X that requires more than T seconds to compute. This will be true for any value of T regardless of how large.