Extreme Value Theorem?

Question

The Extreme Value Theorem says that "If a function is continuous (has no breaks, holes, or jumps) on a closed interval, then it must have both an absolute minimum and an absolute maximum."

Why would a function that is discontinuous not have an absolute minimum and an absolute maximum? Thanks for your help :)

Answer 1

Think of a function which has an asymptote inside the interval (closed or not), where one side goes to infinity, while the other side goes to -infinity. Such a function has an infinite discontinuity where the vertical asymptote is located.



By the way, the closed interval is a necessary condition as well, because f(x) = x on (-1, 1) is a continuous function which does not attain an absolute maximum or absolute minimum on this interval.



I hope this helps!

Answer 2

hi The intermediate cost theorem states that while you're given the values of a carry out on the endpoints of an era [a, b], then the functionality might desire to tackle each well worth in (f(a), f(b)) for x in [a, b] so long because of the fact the function is non-provide up on [a, b]. The propose well worth theorem states that there is a tangent line to a carry out on an era (a, b) whose slope is comparable to the secant line to the function dealing with the climate (a, f(a)) and (b, f(b)) if the carry out is secure on [a, b]. Rolle's Theorem states that if f(a) = f(b) and f(x) is non-provide up on [a, b], then in some unspecified time sooner or later in (a, b), the slope of the tangent line to f(x) is 0. the severe well worth theorem states that if a function is non-provide up on [a, b], then there exist optimal and minimum values for f. i desire this helps!

Answer 3

hi The intermediate fee theorem states that while you're given the values of a carry out on the endpoints of an era [a, b], then the function could desire to attend to each and every properly worth in (f(a), f(b)) for x in [a, b] see you later as a results of fact the perform is non-stop on [a, b]. The mean properly worth theorem states that there is a tangent line to a carry out on an era (a, b) whose slope is comparable to the secant line to the perform dealing with the climate (a, f(a)) and (b, f(b)) if the carry out is secure on [a, b]. Rolle's Theorem states that if f(a) = f(b) and f(x) is non-stop on [a, b], then quicker or later in (a, b), the slope of the tangent line to f(x) is 0. the severe properly worth theorem states that if a perform is non-stop on [a, b], then there exist optimal and minimum values for f. i'm hoping this helps!