Question

Prove the following theorem:

**Theorem.** *Let* *a* *∈* R
*and let* *f* *be a function defined on an
interval centred at* *a**.*

*IF* *f* *is continuous at* *a*
*and* *f*(*a*) *>* 0 *THEN*
*f* *is strictly positive on some interval*

*centred at* *a**.*

Answer #1

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a) prove that the set {x in R:, f(x)>4} is open
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c) give an example of a function F that {x in r, f(x)>4} is
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f is continuous on [a,b]
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bounded and converges to r, and left decreasing sequence and
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limf(a_n)= r= limf(b_n), and f(r)=0.

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trigonometric...

We know that any continuous function f : [a, b] → R is uniformly
continuous on the finite closed interval [a, b]. (i) What is the
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definition is meaningful for functions f : J → R defined on any
interval J ⊂ R.) (ii) Given a differentiable function f : R → R,
prove that if the derivative f ′ is a bounded function on R, then f
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