Answer to Question #158313 in Real Analysis for Cypress

Question

Determine whether these statements are true or false. Explain why.

1) Under the natural ordering, all integers smaller or equal to 0 defines a sequence.

2)The sequence (a)=(n/(n²)) is a monotone sequence. (n∈N)

3) f(x)=sinx is not a polynomial function.

4) The function f(x)=1/x is continuous at x=2

5) All functions that are continuous on (0,1] are bounded.

6) For all functions y=f(x) defined on [0,1], the set f([0,1]) is a bounded set.

7) If y=f(x) is uniformly continuous on a set I, then y=f(x) is bounded on I.

8) If f is a continuous function on a closed interval I, then f(I) is also a closed interval.

9) If limit of f(x)=L when x goes to infinity, then limit of f(1/x)=L when x is goest to 0+ as well.

10) If f is one-to-one, then f^-1(x)=1/f(x)

11) It is impossible for a function to be discontinuous at every number x.

12) Every continuous function on the interval (0,1) has a maximum value and a minimum value on (0,1).

Answer 1

1) true; a sequence is a set with an order in the sense that there is a first element,second element and so on.

2) true

$\{a_n\}=n/n^2=1/n$

$a_{n+1}-a_n=-\frac{1}{n(n+1)}<0$

The sequence is monotonic decreasing.

3) true; a polynomial function is a function that involves only non-negative integer powers or only positive integer exponents of a variable in an equation like the quadratic equation, cubic equation, etc.

4) true; $\displaystyle \lim_{x\to2}f(2)=1/2$

5) false; the function $f: (0,1]\to R$ defined by $f(x)=1/x$ is continuous but not bounded.

6) true

7) true; proof: since it is uniformly continuous, the function is a Lipschitz function. The function is then bounded by the product of two constants, LM, which means that it is bounded.

8) true; the continuous image of a compact set is compact.

9) true

10) false; $f(x)=x^3,$ $f^{-1}(x)=\sqrt[3]{x}$

11) false; In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain.

12) false; it satisfies the extreme value theorem. A continuous function may not have a maximum or minimum if its domain is not confined within a closed interval.