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1.Let f : R → R be a continuous function such that for all x ∈ R we have f(x) ̸∈ Q. Show that f is a constant function. Hint: Assume the contrary: there exist a, b ∈ R with a < b such that f(a) ̸= f(b). Then derive a contradiction.2.. Given a, b, c ∈ R, define functions fa, gb, hc : R → R by fa(x) = { ax if x ∈ Q and 0 if x ̸∈ Q ; gb(x) = { x 2 − x if x ≥ 0 and bx if x < 0 ; hc(x) = { c sin ( 1 x ) if x ̸= 0 and 0 if x = 0 .a.State a necessary and sufficient condition on a so that fa is continuous at 0 (state as: fa is continuous at 0 if and only if P(a) holds). Show the statement. b. State a necessary and sufficient condition on b so that gb is differentiable at 0. Show the statement. c. State a necessary and sufficient condition on c so that lim as x→0 h_c(x) exists. Show the statement.3.This problem is on uniform continuity,a)State the definition of uniform continuity of a function f : R → R, using quantifiers. State the negation of the condition, using quantifiers.b)Given n ∈ N, define a function fn : R → R by f(x) = x n . Show that fn is uniformly continuous on R if and only if n = 1. Hint: In Part 2, the formula: x^n − y^n = (x − y)(x^(n−1) + x^(n−2) y + · · · + xy^(n−2) + y^(n−1) ) is useful.4.Define a function f : R → R by f(x) = e x cos(x).a)Show that f^(4k) (x) = (−4)^k*e^x cos(x), f^(4k+1)(x) = (−4)^k e^x (cos(x) − sin(x)), f (4k+2)(x) = −2(−4)^k e^x sin(x), f^(4k+3)(x) = −2(−4)^k e^x (cos(x) + sin(x)) for all k ∈ {0} ∪ N.b ) Fix M > 0. Show that for all x ∈ [−M, M] and all n ∈ N we have |f (n) (x)| ≤ 2 · 4^n · e^Mc)Show that the Taylor series for f about 0 exists and represents f at all x ∈ R. Find a sequence (c_n)_n∈{0}∪N such that the Taylor series is of the form ∑ from k=0 to ∞ (c_4k*x^4k + c_(4k+1)x^4k+1 + c_(4k+2)x^(4k+2) + c_(4k+3)x^(4k+3)).5.Define a function f : [0, 1] → R by f(x) = e^x . The goal of this problem is to show that f is integrable on [0, 1] and we have ∫ (from 0 to 1)f(x)dx = e − 1 from the definition of the Riemann integral.Given n ∈ N, let Pn to be the even partition of the interval [0, 1] into n subintervals: Pn = { 0 < 1/n < 2/n < · · · < (n − 1)/ n < 1 } . a)Compute the upper sum U(f, Pn).b)Compute the lower sum L(f, Pn).c)Compute limn→∞ U(f, Pn) and limn→∞ L(f, Pn).d)Show that f is integrable on [0, 1] and we have ∫ (from 0 to 1) f(x)dx = e − 1. Hint: As for Part 1,2, the formula ∑(from k=l to m) r^k = r^L − r^(m+1)/( 1 − r ) for r noy equal to 1 is useful. As for Part 3, the formula lim as x→0 (e^x − 1)/x = (e^x )′|_(x=0) = 1 is useful.6.For each of the following, give an example of a function with the prescribed property. You should show that your example indeed satisfies the property. Hint: As for Part 1, 2, the idea for the function fa in Problem 2 is helpful. As for Part 3, the idea for a certain function in a certain homework problem is helpful.a)A function f : R → R that is discontinuous everywhere.b) A function g : R → R that is differentiable at 0 and discontinuous everywhere elsec) A function h: R → R such that both h ′ , h′′ exist on R and h ′′ is discontinuous at 0
Problem 1 (10 points). Let f : R → R be a continuous function such that for all x ∈ R we
have f (x) ̸∈ Q. Show that f is a constant function.
Hint: Assume the contrary: there exist a, b ∈ R with a 0. Show that for all x ∈ [−M, M ] and all n ∈ N we have
|f (n) (x)| ≤ 2 · 4n · eM .
3. (2 points) Show that the Taylor series for f about 0 exists and represents f at all
x ∈ R. Find a sequence (cn )n∈{0}∪N such that the Taylor series is of the form
∞
∑
k=0
(c4k x4k + c4k+1 x4k+1 + c4k+2 x4k+2 + c4k+3 x4k+3 ).
Problem 5 (10 points). Define a function f : [0, 1] → R by
f (x) = ex .
The goal of this problem is to show that f is integrable on [0, 1] and we have
∫ 1
f (x)dx = e − 1
0
from the definition of the Riemann integral.
Given n ∈ N, let Pn to be the even partition of the interval [0, 1] into n subintervals:
{
}
1
2
n−1
Pn = 0
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