3 Limits of functions, ContinuityAfter introducing the basic notions on functions, limits and continuity, wewill go on to Bolzano’s theorem, and the Intermediate Value Theorem (IVT)which follows from it, as well as the Extremal Value Theorem (EVT).3.1 FunctionsDef A function f is a set of ordered pairs (x, y), with x, y ∈ R, such that notwo (ordered pairs) have the same first member.By definition, the second member y is determined by x, one may unam-biguously write y as f(x), where f denotes the assignment x 7→ y. The setof all x (for which f is defined) is called the domain of f, and the set of thecorresponding y is called the image (or range) of f.Notation: f : X → Y , where X is the domain, and Y contains the range.One can plot the ordered pairs {(x, y = f(x))} (defining a function f) inthe Cartesian plane R2= {(x, y) |x, y ∈ R}, and the resulting figure is calledthe graph of f. It will be useful to become aware of the graphs of a numberof standard functions, such as the ones below.Examples:(i) The identity function: f(x) = x;(ii) Constant function: f(x) = c, for all x ∈ R,with c a fixed real number;(iii) Linear function: f = ax + b, for constants a, b;(iv) Polynomial function of degree n ≥ 0: f(x) =∑nj=0ajxj,with a1, . . . , an∈ R, an= 0;(v) Upper semicircle function: f(x) =√r2− x2,X = {x ∈ R | − r ≤ x ≤ r}, r: radius > 0;(vi) The integral part function: f(x) = [x], the largest integer not greaterthan x, X = R, Image(f) = Z.1We will call a function f one-to-one, or injective, iff for every y in therange, there is a unique x such that f(x) = y. The function f : X → Yis said to be onto, or surjective, iff Y is the image of f, i.e., for every y inY , there is an x ∈ X such that y = f (x). Note that the linear functionexample above is injective on X = R iff a = 0, in which case it is onto allof R. Sometimes f is not an injective function on its natural domain X,but becomes one when restricted to a (large enough) subset X1of X. (It isalways injective if restricted to one point, but this is not interesting!) Forexample, the upper semicircle function is injective on {x |0 ≤ x ≤ r}; thegraph of this restriction is a quarter circle of radius r (in the first quadrantof R2).3.2 Open, closed and compact subsets of RBy an interval, we will mean a subset I of R such that if a, b are in I, thenany number x between a and b is in I. Examples are, for a < b ∈ R, theopen interval(a, b) = {x ∈ R |a < x < b},the closed interval[a, b] = {x ∈ R |a ≤ x ≤ b},the half-open intervals[a, b) = {x ∈ R |a ≤ x < b}, (a, b] = {x ∈ R |a < x ≤ b},the infinite open intervals(a, ∞) = {x ∈ R |a < x < ∞},(−∞, b) = {x ∈ R | − ∞ < x < b},and so on.Definition 3.1 Any open interval containing a point a as its midpoint willbe called a standard neighborhood of a.We will usually drop the adjective “standard.”Notation: N(a, r) = {x ∈ R| |x − a| < r} = (a − r, a + r).2By an open set in R, we will mean a subset U which contains a neigh-borhood of every point in U. One can check that open intervals are open, asare arbitrary unions of open intervals and finite intersections of them. Theempty set ∅ and R are open, too.The complement of a set X in R, denoted Xc, is the set R \ X :={z ∈ R |z ∈ X}. Clearly, R and ∅ are complements of each other, while thecomplement of an open interval (a, b) is the union (−∞, a] ∪ [b, ∞).By a closed set in R, we will mean a subset F whose complement Fcis open. Try to verify that closed intervals are closed, as are finite unions ofclosed intervals, and arbitrary intersections of them. The empty set and Rare closed, too. Prove that these two sets are the only subsets of R whichare both open and closed.By a compact set in R, we will mean a subset C of R which is bothclosed and bounded. (Again, when we say “bounded,” we mean it is boundedfrom above and below.) The closed interval [a, b], with a < b in R, is ev-idently compact. The complement of an open interval (a, b) is closed, butnot bounded, so non-compact. And (a, b) itself is bounded but not closed,and so not compact. Compact sets play a very important role in Calculus ofone and several variables, as well as in (higher) Mathematical Analysis andGeometry.3.3 LimitsLet a ∈ R. Assume that f is a function defined on some neighborhood of aexcept possibly at a.Definition 3.2 f has limit A as x → a iff for every neighborhood N1(A)there exists a neighborhood N2(a) such that f (x) ∈ N1(A) if x ∈ N2(a)−{a}.Equivalently, ∀ε > 0, there exists a δ > 0 such that for all x = a, |x −a| < δ,we have |f(x) − A| < ε.Notation: limx→af(x) = A, or f(x) → A as x → a.Theorem 3.3 The following statements are equivalent:(i) limx→af(x) = A3(ii) For every sequence {an}∞n=1⊂ Domain (f), an= a, such that limn→∞an=a, we have limn→∞f(an) = A.Proof (i) ⇒ (ii): Given ε > 0, pick δ > 0 such that |x − a| < δ implies|f(x) − A| < ε, which is possible by (i). Let {an} be a sequence with limita, so that we may choose an N > 0 such that an∈ N(a, δ) for n ≥ N. Thenf(an) ∈ N(A, ε) for n ≥ N, and so (i) holds.(ii) ⇒ (i): Let us prove this by contra-positive, i.e., assume ¬(i) (= Not(i)) and deduce ¬(ii). Pick ε for which δ does not exist, i.e., for every δ > 0,∃x = x(δ) such that |x −a| < δ, but |f(x) −A| ≥ ε. Then picking δ =1nwecan construct a sequence an= x(δn) in N(a,1n) s. t. |f(an) − A| ≥ ε. Thenan→ a but f(an) → A. So (ii) does not hold when (i) doesn’t. 2Right and Left limits: limx→a+, resp. limx→a−.Defined as above except for requiring x ∈ N2(a) ∩ {x > a}, resp. x ∈N2(a)∩{x < a}. This is equivalent to taking sequences an> a, resp. an< a.Clearly, the limit as x → a exists iff the right and left limits both exist andare equal.Examples(1) The limit of a constant function is the same constant.(2) Limit of the identity function limx→ax = a. (In the proof, take δ = ε.)(3) limx→k−[x] = k − 1, limx→k+[x] = k, so limx→k[x] does not exist.Theorem 3.4 Let A = limx→af(x) and B = limx→ag(x). Thenlimx→af(x) + g(x)f(x)g(x)f(x)/g(x)=A + BABA/B if B = 0Proof: Follows from the corresponding statement for sequences.Theorem 3.5 (The squeeze principle) If f(x) ≤ g(x) …