Real Analysis Answers

For which function is given below, there are is a maximum and a minimum values on the given interval?

a) f(x)=x² on (0,1)

b) f(x)=1/x if x=0 on [0,1] , 0 if x=0

c) f(x)=2x+1 on R

d) f(x)=x²+1/x on [1,2]

e) None of the above.

Suppose that y=f(x) is a continous function on x∈[-1,3], and f(-1)=4, f(3)=7. We can conclude that

a) there exists c∈(-1,3) such that f(c)=0

b) there exists c∈(-1,3) such that f(c)=5

c) there exists c∈(4,7) such that f(c)=0

d) there exists c∈(4,7) such that f(c)=5

e) there exists c∈(-1,1) such that f(c)=3

In applying the ε-δ definition of continuity to the function f(x)=8x-5 at x=1, given ε>0, we can choose

a) δ=8/ε

b) δ=ε/8

c) δ=5/ε

d) δ=ε/5

e) δ=8ε

Which of the following functions is uniformly continuous on the given set?

a) f(x)= x³ on [1,infinity)

b) f(x)= x³ on [1,5]

c) f(x)=1/x on (0,1)

d) f(x)=1/x on [0,1]

e) All of the above

A function is uniformly continuous on a set I⊂R if

a) for some ε₀>0 we can find a δ>0 such that if x,y∈I and |x-y|<δ, then |f(x)-f(y)|<ε₀

b) for any ε₀>0 we can find a δ>0 such that if x,y∈I and |x-y|<δ, then |f(x)-f(y)|<ε₀

c) for any ε₀>0 and any δ>0, if x,y∈I and |x-y|<δ₀, then f(x)-f(y)|<ε₀

d) all of the above

e) none of the above

Show that every open interval is an open set

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).

If [x] is the greatest integer not greater than x, then determine the limit of [x ]when x goes to (1/2) is it exists

Find limit superior and limit inferior for the sequence (an)n∈N=((1/n)+(-1)^n)n∈N

Go from 0 to 1 on the x-axis, then back half-way to 1/2, then forward half as far to 3/4, then back as half as far to 5/8, then forward half as far, and so on. Give an explicit formula for your position after the nth step. Where do you end up?