Assume that f is a continuous function on R such that f() = A | $(e)...
5. 6 pt Determine whether the function f(x) is continuous and/or differentiable at x = 1. (x2+1 f(x) = { 12, >1 1 <1
real analysis
hint
13 Suppose fis a continuous function on R', with period 1. Prove that lim Σ f(a)-| f(t) dt 0 for every irrational real number α. Hint: Do it first for f(t)= exp (2nikt), k = 0,±1, ±2, 4.13 Let 2 be the set of functions of form P(t)-Σ_NQC2nikt. The equality holds for functions in . For given ε > 0, there is a P E 2 such that llf-Plloo < ε. Then
Please Answer 135 Below Completely:
Definition Let E-R and f : E-+ R be a function. For some p E E' we say that f is continuous at p if for any ε > 0, there exists a δ > 0 (which depends on ε) such that for any x E E with |x-Pl < δ we have If(x) -f(p)le KE. This is often called the rigorous δ-ε definition of continuity. A couple of things to note about this definition....
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...
Exercise 5. Prove that if f is a continuous and positive function on (0,1], there exists 8 >0 such that f(x) > 8 for any x € [0,1].
check if
e-1/4/ f(x) if x > 0 if x < 0 is differentiable at 0.
11. Let X be a continuous random variable with density function fare-102 for 10 f(1) = lo otherwise where a > 0. What is the probability of X greater than or equal to the mode of X?
-. The function f has derivatives of all orders for -1 << < 1. The derivatives of f satisfy the following conditions: f(0) = 0 f(0) = 1 f(n+1) f(n)(0) for all n > 1 The Maclaurin series for f converges to f(x) for all 3 <1. (a) (5 points) Write the first four nonzero terms of the Maclaurin series for f. (b) (5 points) Determine whether the Maclaurin series described in part(a) converges abso- lutely, converges conditionally, or diverges...
3. Let f : [0, 1] → R be uniformly continuous, so that for every e > 0, there exists >0 such that |x – y < =\f(x) – f(y)] < e for every x, y € [0, 1]. The graph of f is the set Gf = {(x, f(x)) : € [0, 1]}. Show that Gf has measure zero (9 points).
PROVE:
4. If f : R → R is a strictly increasing function, f(0) = 0, a > 0 and b > 0, then