Analysis Flashcards
Lemma; Prove that f(x) = sqrt(x) is continous
Lemma; If f,g: E to R are continuous at c then f+g is continuous at c
Lemma; Suppose that f,g: E to R are continuous at c in E. The the function |f| is continuous
Show that the function
f(x) = 1/q for rationals when x = p/q
0 for irrationals
is discontinous at every rational
Theorem: The function f: R to [-1,1] given by f(x)=sinx is continuous
Show that x^2 = 2 has a root in the interval (0,2)
Consider f(x) = x^2 -2
f(0)= -2
f(2) = 2
so by IVT there is a root
Proposition; Any odd degree polynomial has at least one real root
Correction: Take the x* and x* to be 2A/a2n+1 and -2A/a2n+1
Lemma; Any continuous function f: [a,b] to [a,b] has a fixed point ie there is an x* in [a,b] such that f(x*) = x*
Prove the Extreme Value Theorem
Lemma; The interval (a,b) is open
Lemma; Suppose that A and B are open subsets. Then A u B and A n B are open
Lemma; The interval R[a,b] is closed
We have R[a,b] = (-inf,a) u (b,inf). This is a union of two open sets, so its open
Lemma; A subset A of R is closed if and only if (an) in A with an tends to a implies that a is in A
Lemma; A function f:R to R is continuous if and only if f-1U is open for every open set U in R
State and prove the sandwich rule for continuous limits
State and prove the continuous limits and composition proposition.
Show that if f(x) = sinx then f’(x) = cosx
Lemma; If f is differentiable at x0 then f is continuous at x0
Lemma; Suppose that f,g: (a,b) to R are differentiable at x0 then f + g is differentiable at x0
Lemma; Suppose that f,g: (a,b) to R are differentiable at x0 then fg is differentiable at x0
Lemma; If g is differentiable at x0 and g(x0) doesnt equal 0 then 1/g is differentiable at x0
Prove the Caratheodory formulation of differentiation
State and prove the chain rule
Prove the derivative of inverses theorem
State and prove L’Hopitals rule
Lemma; Let f:(a,b) to R. Suppose that x0 in [a,b] is a local maximum or minimum of f, and that f is differentiable. Then either
i) x0 is an endpoint
ii) f’(x0) = 0
Prove Rolles Theorem
Lemma; An nth degree polynomial can have at leat n distinct real roots
Let P be a polynomial of degree n with at least n+1 distinct roots. Then by noting that P is infinitely differentiable and that all derivatives are continuous we can repeatedly apply Rolles Theorem, Then P(1)(x) has at least n distinct roots,….., P(n)(x) has at least 1 root. But P(n) is a non-zero constant, so this is impossible
Prove Taylors Theorem
Lemma; Any nth degree polynomial P is completely determined by its first n derivatives at any point a in R
