Exercise 1. This exercise builds on the method used to prove that if a function differetiable at ...
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint: Use the definition of derivative and then use mean value theorem) 2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) =...
Real analysis 10 11 12 13 please (r 2 4.1 Limit of Function 129 se f: E → R, p is a limit point of E, and limf(x)-L. Prove that lim)ILI. h If, in addition, )o for all x E E, prove that lim b. Prove that lim (f(x))"-L" for each n E N. ethe limit theorems, examples, and previous exercises to find each of the following limits. State which theo- rems, examples, or exercises are used in each case....
PLEASE ANSWER ALL! SHOWS STEPS 2. (a) Prove by using the definition of convergence only, without using limit theo- (b) Prove by using the definition of continuity, or by using the є_ó property, that 3. Let f be a twice differentiable function defined on the closed interval [0, 1]. Suppose rems, that if (S) is a sequence converging to s, then lim, 10 2 f (x) is a continuous function on R r,s,t e [0,1] are defined so that r...
Exercise 3.1.12: Prove Proposition 3.1.17. Exercise 3.1.13: Suppose SCR and c is a cluster point of S. Suppose : S R is bounded. Show that there exists a sequence {x} with X, ES\{c} and lim X e such thar S(x)} converges. and g such thal 2 2 asli and 8 ) Las y C2, bulg 1)) does not go lo L as is, find x → Exercise 3.1.15: Show that the condition of being a cluster point is necessary to...
3. Suppose f : [0,) + R is a continuous function and that L limf(x) exists is a real number). Prove that f is uniformly continuous on (0,.). Suggestion: Let e > 0. Write out what the condition L = lim,+ f(t) means for this e: there erists M > 0 such that... Also write out what you are trying to prove about this e in this problem. Note that f is uniformly continuous on (0.M +1] because this is...
hint This exercise 5 to use the definition of Riemann integral F. Let f : [a, b] → R be a bounded function. Suppose there exist a sequence of partitions {Pk} of [a, b] such that lim (U(Pk, f) – L (Pk,f)) = 0. k20 Show that f is Riemann integrable and that Så f = lim (U(P«, f)) = lim (L (Pk,f)). k- k0 1,0 < x <1 - Suppose f : [-1, 1] → R is defined as...
have a test tom, could really use some help with these questions please. please help Question 0.4. Find the limit (it it exists) of lim(2r +l -3) 11η Question 0.5. For what values of the constant c is the function f continuous everyuhere ca2 +2r when x < 2 3-cr otherwise -z has a root in (0.1). Question 0.6. Use intermediate value theorem to show Question 0.7. Find lim,→x (V9pH- 7. Find limgtoo (V9x2 +x-3r Question. Prove f(x) = (x...
real analysis 1,3,8,11,12 please 4.4.3 4.4.11a Limits and Continuity 4 Chapter Remark: In the statement of Theorem 4.4.12 we assumed that f was tone and continuous on the interval I. The fact that f is either stric tric. strictly decreasing on / implies that f is one-to-one on t one-to-one and continuous on an interval 1, then as a consequence of the value theorem the function f is strictly monotone on I (Exercise 15). This false if either f is...
(a) Suppose that lim x→c f(x) = L > 0. Prove that there exists a δ > 0 such that if 0 < |x − c| < δ, then f(x) > 0. (b) Use Part (a) and the Heine-Borel Theorem to prove that if is continuous on [a, b] and f(x) > 0 for all x ∈ [a, b], then there exists an " > 0 such that f(x) ≥ " for all x ∈ [a, b]. = (a) Suppose...
4 Suppose f : (0,0) → (0,x), is a differentiable function satisfying f(a +b)-f(a)fb), for all a,b>0 Moreover, assume that f(0)1 (a) Prove that there exists λ (not necessarily positive) such that f(r) = e-Ar, for all r. Hint Find and solve a proper differential equation. (b) Suppose that X is a continuous random variable, with P(X>ab)-P(>a)P(X> b), for all a, b e (0, oo). Prove that X is exponentially distributed