Real Analysis Answers

Show that the linear combination of two functions of bounded variation is also of bounded variation. Is the product of two such functions also of bounded variation? 

Define f(x) = sinx on [0, 2pi]. Find two increasing functions h and g for which f = h — g on 

[0, 2pi]. 

Construct an example of a function  f(x) that is defined at every point in a closed interval and whose values at the end points have opposite signs but still f(x)=0 has no solution in the interval.

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).