Give an example of a bounded function that is defined on R, continuous at each point except the points 1 and -1, and has neither a removable discontinuity nor a jump discontinuity at 1 and-1.

Let X be the set {5, -6, 2}. For the first three parts of this problem you are asked to define a function f: X→ X so that the relation un w w = f(u) satisfies each of the following conditions. (a) is reflexive f(x) (b) ~ is symmetric -6 f(x) (c)~ is transitive f(x) | 2)

Let h(g₁, 92) be a continuous function through a nonempty, closed and bounded set S. Are these conditions necessary, sufficient, necessary and sufficient or neither necessary nor sufficient conditions to ensure that there exists points (a, b) and (c, d) in S where h has a minimum and a maximum value, respectively? Necessary and sufficient Sufficient, but not necessary Neither necessary nor sufficient Necessary, but not sufficient None of the above No answer

Suppose that f is a function with f(2) > 0 and f1(2) = 0. If f is a local minimum of f, what do we know about f 11(2)?