MA104 EXAM IIFIRST: Write your name and circle, in INK, whether you choose the ORIGINALgrading option, or the REVISED grading option. The original counts all threemidterms equally, each 25 percent of the total score. The revised option cancelsthe first midterm, but weights the second two at 37.5 percent of the total score.THEN: You can begin reviewing the next page and jotting down any definitions ortheorems you think will help. We will all start the test at 4:15, and stop at 6:00.12THIS PAGE IS FOR REFERENCEDefinition: f : X → Y is continuous at x ∈ X if, for all sequences (xn) ⊆ X takenfrom X, with xn→ x, we also have f(xn) → f (x) ∈ Y .Definition: f : X, d → Y, ρ is uniformly continuous on X if, ∀ε > 0, ∃δ > 0, suchthat whenever x1, x2∈ X, with d(x1, x2) < δ, we also have ρ(f (x1), f (x2)) < ε.Definition: We say the sequence of functions fn: X → R converges uniformly tof : X → R if the sequence of real numbers (d∞(fn, f )) ⊆ R satisfiessup{|fn(x) − f (x)| : x ∈ X} = d∞(fn, f ) → 0 as n → ∞.This is also the definition of convergence in the metric space Cb(X), d∞, whereCb(X) := {f : X → R | f is bounded and continuous on X}.3Problem 1:Let f : R → R be given by f (x) = 0 if x is irrational, and f (x) = 1 − x2if x ∈ Q.Determine completely, with proof, the following sets:C := {x ∈ R : f is continuous at x}N := {x ∈ R : f is not continuous at x}4Problem 2: Select (i) or (ii).(i) Suppose a, b ∈ R with a < b. Show that if f : (a, b) → R is a uniformlycontinuous function on (a, b), it is bounded on (a, b).Is it true that all bounded continuous functions on bounded open intervals areuniformly continuous on their domains? Justify your response with a proof orcounterexample.(ii) Show that the function f : (0, ∞) → R given by f (x) =sin(x)xis uniformlycontinuous on its domain.You may assume the usual facts from calculus regarding the sin(·) function.5Problem 3:Consider the sequence of functions fn(x) =11+xn, fn: [0, ∞) → R. Using arrownotation:fn=x 7→11 + xnCompute the pointwise limit functionf(x) = limn→∞fn(x)for each x ∈ [0, ∞).Do the fnconverge uniformly to f on [0, 1]?Is there a C > 0 such that fnconverges uniformly to f on [C , ∞)?Justify both of the responses to the questions. You do not have to show work forthe computation.6Problem 4:Supp ose that f : R → R is continuous, and f (a)f (b) < 0 for some pair of realnumbers a < b. Show that there is an x ∈ R with f (x) = 0.7Problem 5:Briefly justify your responses in the following questions, with either an example ora reason why an example does not exist. You may select familiar functions fromfreshman calc ulus such as sin, arctan, etc. and assume their familiar properties inyour examples. There are 7 total. Your score will be max(4(n − 2), 0) where n isthe number, out of 7, that you answer correctly.Does there exist a continuous function that maps...(a) ... R onto exactly [0, 1]?(b) ... [0, 1] onto all of R?(c) ... (0, 1) onto exactly R − {0}?(d) ... (0, 1] onto all of R?(e) ... R onto exactly (0, 1]?(f) ... (0, 1) onto exactly [0, 1]?(g) ... [0, 1] onto exactly (0, 1)?Recall that we say ‘f maps E onto F ’ if E ⊆ dom(f) and f(E) = F , as sets. Ihave added the words ‘exactly’ and ‘all of’ to help make the meaning clearer.Hint: There is nothing special about the numbers 0 and 1, other than 0 < 1.8Problem 6:Let g : R → R be a given bounded uniformly continuous function. For each a ∈ R,consider the functionga: R → Rgiven byx 7→ g(a + x).In other words, ga(x) = g(a + x). This enables us to define a function fromR → {bounded continuous functions on R}viaa 7→ ga.Call this function F . Rewriting,F : R → Cb(R)is given byF (a) = ga= [x 7→ g(a + x)].Show that F is uniformly continuous as a map of metric spaces, where the metriconCb(R) = {f : R → R | f is bounded and continuous}is given byd∞(f, g) = sup{|f (x) − g(x)| : x ∈