MATH 521 Exam 3

Riemann Integral

Approximation of area under curve by partitions Min/Max exist by EVT L(f,P) = ∑ mi (xi+1 - xi) ≤ area under graph U(f,P) = ∑ Mi (xi+1 - xi) ≥ area under graph As partition gets refined (increases), the [xi, xi+1] intervals get smaller and CV to a (NIP: each greater partition is nested)

Differentiability and Continuity

Differentiability implies continuity (not cont. => not diff) But not v.v. (consider |x| function - f'(0) DNE as slope is different approaching from either side) Taking derivative just shows derivatives continuity, but nothing on differentiability

Differentiability

Differentiable at c iff lim x->c f(x)-f(c)/x-c exists <=> lim h->0 f(c+h)-f(c)/h exists For all ε>0 there exists ∂(ε,c)>0 s.t. |x-c|<∂ => |f(x)-f(c)/x-c - L|<ε For all xn where xn->c => |f(xn)-f(c)/xn-c - L|<ε If differentiable, derivative exists at c Algebraic differentiability applies f diff at all c on (a,b) means that f' function exists on (a,b)

Uniform Continuity

For all ε > 0 there exists a ∂(ε)>0 s.t. for all x, y ∈ A, |x−y|<∂ => |f(x)−f(y)|<ε Domain based, only dependent on ε (not c) f continuous on compact set => f uniformly continuous Same "box" applies to full graph; impossible w/ asymptote

Uniform CV of a Sequence of Functions

For all ε>0 there exists N(ε) s.t. for all n > N(ε) |fn(x) - f(x)| < ε for all x in D (where f(x) is the pointwise limit - uniform if g'(x) = lim g'n for all x; uniform if Mn = sup |fn(x) - f(x)| CV to 0) Independent of x, implies pointwise CV (same for all x) Cauchy (w/out explicit limit): for all ε>0 there exists N(ε) with n,m natural s.t. for all n,m > N(ε) |fn(x) - fm(x)| < ε for all x in D

Fundamental Theorem of Calculus

For an f: [a,b] and f bounded on [a,b]: If there exists F s.t. F'(x) = f(x) for all x in (a,b), then AUC = ∫ab f = F(b)-F(a) If f is continuous and integrable at c on (a,b), then x->G(x) = ∫ax f(t)dt is diff at c and G'(c) = f(c) (G is the antiderivative of f where f is cont.)

Theorems using Differentiability

If differentiable, f has an extremum at some c on (a,b) s.t. f'(c)=0 Rolle's: if f is cont. on [a,b], f is diff on (a,b), and f(a) = f(b), then there exists a c in (a,b) s.t. f'(c)=0 (continuous, differentiable, endpoints are equal = has a max/min by EVT) Mean Value: if f is cont. on [a,b] and f is diff on (a,b), then there exists a c in (a,b) s.t. f'(c) = f(b)-f(a) / b-a (at some point, tangent's slope = slope between endpoints) - can go btwn UCV and MVT easily MVT => Rolle's if f(a) = f(b) Darboux: f diff on [a,b] and f'(a) < β < f'(b) then there exists c in (a,b) s.t. f'(c) = β (can use g(x) = h(x) - x to solve)

Relationship between fn and f

If fn: (a,b) -> R, fn continuous on (a,b), and fn CV uniformly to f on (a,b), f is continuous on (a,b) (ε/3 proof) fk cont. on (a,b), then Sn cont. on (a,b) Sn CV uniformly on (a,b) then S = limn->∞ Sn cont. on (a,b) (evaluate using Cauchy CV of partial sums - Sn/Sm tilde) fn: (a,b) -> R, fn CV pointwise to f on (a,b), fn diff on (a,b), f'n CV uniformly to g on (a,b), then f is diff on (a,b) and f' = g (derivative of limits = limit of derivatives)

Riemann Integrals and Sequences of Functions

If fn:[a,b] -> R, fn R integrable, and fn CV uniformly to f on [a,b], then f is R integrable on [a,b]

f is Riemann integrable if...

L(f) = supL(f,P), U(f) = infU(f,P) f is R integrable on [a,b] if f's range is bounded and if U(f) = L(f) = ∫ba f (U(f) - L(f) < ε) f is R integrable on [a,b] if f is continuous on [a,b] For improper integrals, a = lim v->∞ ∫∞a f(t) dt (taking limit as v goes to infinity) Similarly if range is unbounded, can take limit as variable for either endpoint approaches that endpoint

General Limit info

Limit can exist even if f isn't continuous at point c Limit DNE if + limit doesn't equal - limit Limit DNE if xn, yn ≠ c; lim xn = lim yn = c, but limf(xn) ≠ limf(yn) c doesn't have to be in the domain but it has to be a limit point at least Algebraic limit applies

Theorem on Limits

TFAE: (1) lim x->c f(x) = L (2) for all ε > 0 there exists a ∂(ε, c) > 0 s.t. |x-c| < ∂ => |f(x) - L| < ε (3) For all xn and xn->c, xn ≠ c => f(xn)->L

Riemann Integrals and Discontinuities

f bounded on [a,b], f cont. at all points of [a,b] except finitely/countably many => f is R integrable Want U(f,P) - L(f,P) < ε, so pick 2∂M (area of discontinuity) < ε/3 s.t. U(f,P) - L(f,P) ≤ (U(f1,P1) - L(f1,P1)) + (U(f2,P2) - L(f2,P2)) + 2∂M =ε Infinitely many discontinuities => not R integrable f bounded on [a,b] and integrable on [b,c] for c in [a,b] => f is integrable on [a,b]

Theorem on Continuity

f continuous at c iff limf(x) = f(c) as x->c f continuous at c if for all ε>0 there exists ∂(ε,c)>0; |x-c|<∂ => |f(x)-f(c)|<ε If c is isolated, lim x->c f(x) DNE but still continuous Not continuous at c if there exists xn in A where xn->c but not f(xn)->f(c) Algebraic continuity applies

Continuity and Compactness

f continuous on [a,b] then f([a,b]) = [m,M] k compact -> f(k) compact m=Minf M=Maxf exist (EVT) y=b intersects graph at least 1 time (IVT) for all b in compact set - can use to prove roots Prove by f([a,b]) bounded, closed, connected based on compactness of [a,b]

Pointwise CV of a Sequence of Functions

fn CV pointwise on D if lim n->∞ fn(x)=f(x) for all x in D For all x in D, fn(x) -> f(x) For all x in D and all ε>0, there exists N(ε,x) s.t. for all n>N |fn(x) - f(x)| < ε As fn(x) is a real number, we can use ALT, OLT, etc. fn can be continuous without its limit f being continuous So, not powerful enough to keep cont./diff. at limit

M test for uniform CV of function sequence's sum

∑ fk(x) = Sn(x), fk defined on (a, b); Sn CV uniformly on (a,b) if |fk(x)| ≤ Mk for all x in (a,b) and ∑ Mk < ∞ (Mk CV)

Nowhere Differentiable Continuous Functions

∑ g((2^k)x)/2^k is continuous and CV by geometric series, but is not diff at all {p/2^k; p being an integer} for a fixed x Summation is continuous at it is the finite sum of continuous functions