Editing Openai/6935580c-34e4-8001-9136-7bbdf1499790 (section)

=== You wrote: “if it is discontinuous, take the least sss for which it is discontinuous and prove as below its continuity from continuity of lesser sss.” That argument fails for a simple reason: the set of discontinuities of a function on [0,t][0,t][0,t] need not have a minimum. Typical counterexamples: ===
'' there can be a sequence of discontinuities sn↓s∗s_n\downarrow s_''sn↓s∗ and s∗s_*s∗ itself may be a continuity point (so no minimum), or
* the discontinuities can accumulate at 000, so there is no least positive discontinuity.

So you cannot rely on a “least sss” unless you can prove a priori that the set of discontinuities, if nonempty, contains a minimum — and nothing in your premises guarantees that. In short: the real line is not well-ordered, and continuity/discontinuity sets need not have minima.