Consider the function f:R + R defined by if x is rational f(x) = if x is irrational. Find all c € R at which f is continuous. C
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...
5. Define f:R → R by f(x) = ax + b where b = 0 Is f a linear transformation? Justify your answer.
*14. Let A be an n x n matrix. Define f:R" R by f(x) = Ax.x = x'AX. (a) Show that f is differentiable and Df (a)h = Aah + Ah a. (b) Deduce that when A is symmetric, Df(a)h = 2Aa . h. 15. Let a € R", 8 >0, and suppose f: B(a, 8) - R is differentiable at a. Suppose f(a) f(x)
Let r be any rational number and define L = { x in Q: x < r }, the set of rational numbers less than r. Show that L is a Dedekind cut by proving the following properties: A. There exists a rational number x in L and there exists a rational number y not in L. ( This proves L is nonempty and L is not equal to Q) B. If x in L, then there exists z in...
(6) Let fel ), where is Lebesgue measure on R. Define F:R → R by F(x) = f' f(t) dx. (a) Prove that F is a continuous function. (b) Prove that F is uniformly continuous on R. (Note that R is not compact.)
Problem 1: Determine whether the statement is true or false. If the statement is true, then prove it. Otherwise, provide a counterexample. (a) If a continuous function f:R +R is bounded, then f'(2) exists for all x. (b) Suppose f.g are two functions on an interval (a, b). If both f + g and f - g are differentiable on (a, b), then both f and g are differentiable on (a,b). Problem 2: Define functions f,g: RR by: x sin(-),...
7. Consider the function f:R + R defined by f(x) = x < 0, 3 > 0. e-1/x2, Prove that f is differentiable of all orders and that f(n)(0) = 0 for all n e N. Conclude that f does not have a convergent power series expansion En Anx" for x near the origin. [We will see later in this class that this is impossible for holomorphic functions, namely being (complex) differentiable implies that there is always a convergent power...
4. Define f(z) ={z. (Lia z, İftE [0, 1] rational; -z, if x [0,1 irrational 1 f(x) = if z E (, i] rational; Prove that the function f is not integrable on
. Let g(x): 0 if x [0, 1] is rational and g(x) 1/x if x [0, 1] is irrational. Explain why g R[0, 1]. However, show that there exists a sequence (P") of tagged partitions of [a, b] such that |Pl 0 and lim,S(g; P) exists.