- #1
xsw001
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Let f:[a,b]->R be continuous and one-to-one such that f(a)<f(b).
Let a<c<b. Prove that f(a)<f(c)<f(b)
My first instinct is to apply intermediate value theorem. Let me know whether my proof makes sense or not.
Proof:
Since f:[a,b]->R is continuous and one-to-one.
Therefore f is strictly increasing function.
Suppose a<c<b
According to Intermediate Value Theorem
There exists f(c) such that f(a)<f(c)<f(b)
Let a<c<b. Prove that f(a)<f(c)<f(b)
My first instinct is to apply intermediate value theorem. Let me know whether my proof makes sense or not.
Proof:
Since f:[a,b]->R is continuous and one-to-one.
Therefore f is strictly increasing function.
Suppose a<c<b
According to Intermediate Value Theorem
There exists f(c) such that f(a)<f(c)<f(b)
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