Math for Non-Geeks/ Proving discontinuity

oben Overview Recall that whenever we want to prove the negation of a statement about some elements of a set, we need to show that there as at least one element in that set for which the statement is not true. So, in order to prove the discontinuity of a function, all you have to show is that the function has (at least) one point of discontinuity. There are several methods available for proving the existence of a point of discontinuity: Sequence criterion: Show that the function doesn't fulfill the sequence criterion at one particular point Considering the left- and right-sided limits: Calculate the left-sided and right-sided limits of the function at a particular point. If either one of these limits doesn't exist, or if the limits are different, then the function is discontinuous at that point. Epsilon-Delta Criterion: Show that the function doesn't fulfill the epsilon-delta criterion at a particular point. Sequence Criterion Sequential_definition_of_continuity Review: Sequence Criterion Sequence criterion of continuity for a single argument Sketch of the proof Sketch of proof:discontinuity Example exercises Exercise:Discontinuity of the topological sine function Epsilon-Delta Criterion Epsilon-delta definition of continuity Review: Epsilon-Delta Criterion Definition:discontinuity General proof structure The Epsilon-Delta criterion of discontinuity can be formulated in predicate logic as follows: From here we get a scheme that allows us to prove the discontinuity of a function using the delta-epsilon criterion: Example exercise Problem:Topological sine function Exercises Epsilon-delta criterion: Sign function Problem:Discontinuity of the sign function unten