3. Suppose f:[a, b]→ Ris a monotonically increasing function. If f assumes every value between f(a) and f(b) at least once, then show that f is continuous on [a, b]. Suppose f:RR is a continuous function on R. Let a, b eR be such that a < b, then prove that f(Ja, b[ ) is an open interval in R, where Ja, b[ denotes an open interval in R.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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3. Supposc f:[a, b]→ R is a monotonically increasing function. If f assumes every value between
f(a) and f(b) at least once, then show that f is continuous on [a, b].
Suppose f: R →R is a continuous function on R. Let a, b ER be such that a < b, then prove that
f(Ja, b[ ) is an open interval in R, where Ja, b[ denotes an open interval in R.
Transcribed Image Text:3. Supposc f:[a, b]→ R is a monotonically increasing function. If f assumes every value between f(a) and f(b) at least once, then show that f is continuous on [a, b]. Suppose f: R →R is a continuous function on R. Let a, b ER be such that a < b, then prove that f(Ja, b[ ) is an open interval in R, where Ja, b[ denotes an open interval in R.
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