Functions, inverse images, and continuity

A function assigns exactly one output point to every input point. To study its interaction with observation, also ask about target events: which inputs land inside a specified subset of the target? This question runs backward through the point arrows. That set of inputs is the inverse image of U. It is not an inverse function. Several inputs may have the same output, and the inverse image is still defined even when no inverse function exists. On a two-point target, let every input map to one. The inverse image of the singleton containing one is the whole source. The inverse image of an event missing one is empty. Both sets are open in any source topology. A function is continuous when the inverse image of every open target event is open in the source. This definition concerns events, not whether the arrows look smooth or endpoints look near each other in a drawing. Use separate two-point source and target spaces. Each has opens empty, the singleton one, and the whole two-point carrier; the singleton zero is not open. Consider negation as a point function, exchanging zero and one. Test continuity by pulling back the target open singleton one. Under negation, only input zero lands at one. The inverse image is the singleton containing zero, which is not source-open. One failed target-open test disproves continuity; the map's bijectivity does not repair that failure. A selected formula is a true-or-false question on valuations. Its true-row region is the set where the answer is one. These regions generate the observation opens using unions and finite intersections, together with the empty and whole sets. Each selected formula defines a bit-valued function: it sends a valuation to that formula's truth value. Give its two-point target the Sierpinski topology. The inverse image of the target open singleton one is exactly the selected formula's truth region. That truth region is a designated subbasic open in the observation topology. The other target opens pull back to empty and the whole carrier. Thus every selected coordinate map is continuous by direct checking of all target opens. It is enough to check inverse images of target subbasic opens. Inverse images preserve intersections and unions, so once those generators pull back to opens, all their finite intersections and arbitrary unions do too. Continuity is a statement about a function together with two topologies. Changing either observation menu can change whether the inverse-image test passes. A point-arrow diagram without its source and target opens does not establish continuity. On the positive two-point spaces, compare the constant-one map with negation. What is the inverse image of the target singleton one for each, and which map is continuous? The constant-one preimage is the whole source, which is open. Negation gives the non-open singleton zero. The constant map passes all open tests and is continuous; negation fails this one and is not. Inverse images translate target events into source conditions. Continuity says every observable open target condition remains observable after that translation. Next, the finite information orders will let us check the same property through preservation of comparisons.