3. Suppose f : [0,) + R is a continuous function and that L limf(x) exists...
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) =...
3. (a) Suppose f : (a, b) + R is differentiable, and there exists M E R such that If'(x) < M for all x € (a, b). Prove that f is uniformly continuous on (a, b). (b) Let f : [0, 1] → [0, 1] be a continuous function. Prove that there exists a point pe [0, 1] with f(p) = p.
Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some zo R, then f has the maximum on R, that is, there exists an M R such that f(x) < f(xM) for al E R.
Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some zo R, then f has the maximum on R, that is, there exists...
1. Let f:R → R be the function defined as: 32 0 if x is rational if x is irrational Prove that lim -70 f(x) = 0. Prove that limc f(x) does not exist for every real number c + 0. 2. Let f:R + R be a continuous function such that f(0) = 0 and f(2) = 0. Prove that there exists a real number c such that f(c+1) = f(c). 3 Let f. (a,b) R be a function...
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....
Problem 1. Suppose that f:(a,b) + R is a continuous function and there exists a point p e (a, b) such that f' exists and is bounded on (a,b) {p}. Prove that f is uniformly continuous on (a,b).
(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...
Let f be defined on an open interval I containing a point a (1) Prove that if f is differentiable on I and f"(a) exists, then lim h-+0 (a 2 h2 (2) Prove that if f is continuous at a and there exist constants α and β such that the limit L := lim h2 exists, then f(a)-α and f'(a)-β. Does f"(a) exist and equal to 2L?
Let f be defined on an open interval I containing a point a...
3) Prove that there exists f : R → R non-negative and continuous such that f € L'OR : dm) ( i.e. SR \f|dm <00) and lim sup f(x) = ∞. 2-0
For each n E N, define a function fn A - R. Suppose that each function fn is uniformly continuous. Moreover, suppose there is a function f : A R such that for all є 0, there exists a N, and for all x E A, we have lÍs(x)-f(x)|く for all n > N. Then f is uniformly continuous. Note: We could say that the "sequence of functions" f "converges to the function" f. These are not defined terms for...