Determine the Validity of the following statements... (A) Continuous Functions t

Question

Determine the Validity of the following statements...

(A) Continuous Functions take bounded open intervals to boundedopen intervals

(B) Continuous Functions take bounded open intervals to opensets

(C) Continuous Functions take bounded closed intervals to boundedclosed intervals.

I know I have to somehow use the Intermediate Value Theorem

Explanation / Answer

(A) + (B): This is false. The the function f:R->R defined byf(x)=x2 and the open interval (-1,1). The image under fof this open interval is the half open interval [0,1). NB: For (A) you could ask what happens with the bounded part. Thisdepends a little on the domain of your function. If you takef:(0,1)->R defined by f(x)=1/x, then the image of (0,1) underthis function is unbounded. If the domain of your function is thewhole real line, then a bounded open interval is contained in abounded closed interval. This bounded closed interval has a boundedimage by (C): (C): This is true. Let f be a continuous real valued functiondefined on the closed and bounded interval D. Then f[D] is again abounded and closed interval. You can see this in 2 steps: first weshow that f[D] is an interval. For this part apply the intermediatevalue theorem. This theorem says that the image of an interval isan interval, so f[D] is an interval* (an interval is anything ofthe form (a,b), [a,b], (a,b] or [a,b), where the value isalso allowed for a or b). Next we need to show that f[D] is closed and bounded. By theHeine-Borel Theorem, the interval D is compact (since it is closedand bounded. The continuous image of a compact set is compact, sof[D] is compact. Now apply the HB-Theorem again to conclude thatf[D] is closed and bounded. *The IVT theorem really says that if c nd d belong to f[D] andc

Related Questions

Navigate

• Browse (All)
• Browse D
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.