Math for Non-Geeks/ Proving continuity

oben Overview There are several methods for proving continuity: Concatenation Theorems: If the function can be written as a concatenation of continuous functions, it's continuous by the Concatenation Theorems. Using the local nature of continuity: If a function looks like another well-known continuous function in a small neighborhood of a point, then it must also be continuous in this point. Considering the left- and right-sided limits: If one can show that the left- and right-sided limits of a function are the same in some point, then the function is continuous in this point. Showing the sequence criterion: Using the sequence criterion means that the limit can be pulled into the function, i.e. we can consider the limit of the arguments. For a sequence of arguments with limit value it must hold . Showing the epsilon-delta criterion: For every it must be show that there exists some such that for all arguments with a distance smaller than from the point , the inequality is satisfied. Composition of continuous functions Composition of continuous functions General proof sketch Sketch of the proof Example Problem Problem:Epsilon-delta proof of continuity for a square root function Using the Local Nature of Continuity If applicable, one can use the fact that continuity is a local property. Namely, when a function looks the same as another function in a certain point , i.e. on some small neighborhood of , and we know to be continuous, then must also be continuous at . For example, consider the function with for positive and for negative . Now we choose some random positive number . In a sufficiently small neighborhood of , is constant : center|450px|The function f is constantly 1 in a sufficiently small neighborhood of a positive argument. Since constant functions are continuous, is also continuous at the point . Similarly, one can show that is also locally constant for negative number. Then is also continuous for negative numbers, and is therefore continuous for all real numbers. In the proof we write: "For every there exists a neighborhood around where is either constant or constant . Since constant functions are continuous, must also be continuous at the point . Therefore, is continuous“. Such an argument can often be applied to functions that are defined by a case differential. Our function is a good example of that. In conclusion, it's defined as: However, the local nature of continuity can not be used as an argument for every function defined by a case differential! Let's consider the following function : For all points that are not zero we can construct a proof like we've done earlier in this chapter to show that the function is continuous at these points. However, at the point we have to construct a different argument. For example, here we could consider the left- and right-sided limits. Sequence Criterion Sequential definition of continuity Review: Sequence Criterion Sequence criterion of continuity General Proof Structure Beweisskizze:Stetigkeit Example Problem Continuity of quadratic functions Epsilon-Delta Criterion Epsilon-Delta-Kriterium Epsilon-delta definition of continuity Review: Epsilon-Delta Criterion Definition General Proof Structure In mathematical quantifiers, the epsilon-delta definition of continuity of a function at the point reads: This technical method of writing the claim specifies the general proof structure for proving continuity using the epsilon-delta criterion: Example Problem and General Procedure Problem:Quadratic function Example Problems Sequence Criterion: Absolute Value Function Exercise:Continuity of the absolute function Epsilon-Delta Criterion: Linear Function Problem:Continuity of a linear function Epsilon-Delta Criterion: Concatenated Absolute Value Function Problem:Concatenated absolute function Epsilon-Delta Criterion: Hyperbola Exercise:Hyperbola Epsilon-Delta Criterion: Concatenated Root Function Problem:Epsilon-delta proof of continuity for a square root function unten