![f) -1 increasing on (011] 乂 } Sudt ne minima (ie. f à unbounded eloo) L-e יג thatdotu not have a mininvm](http://img.homeworklib.com/questions/21e10400-923b-11eb-aa0f-154d0e18b80d.png?x-oss-process=image/resize,w_560)
Find a continuous function f: (0,1] → R that does not have a minimum.
- Let V be the vector space of continuous functions defined f : [0,1] → R and a : [0, 1] →R a positive continuous function. Let < f, g >a= Soa(x)f(x)g(x)dx. a) Prove that <, >a defines an inner product in V. b) For f,gE V let < f,g >= So f(x)g(x)dx. Prove that {xn} is a Cauchy sequence in the metric defined by <, >a if and only if it a Cauchy sequence in the metric defined by...
(7) Let R= {f [0,1] - R | f continuous} be the ring of all continuous functions from the interval [0,1] to the real numbers. (a) For cE [0, 1, prove that Me := {feR | f(c) = 0} is a maximal ideal of R. Hint: consider the evaluation map ec- (b) Show that if M is any maximal ideal of R then there exists a cE [0,1 such that M = Me. Hint: show that any maximal ideal M...
CELLERIA (b) (6) Suppose f is continuous. If f is restricted to [0,1] but contains no critical values in that range, what can you say, if anything, about the extreme values off on (0,1)? (u) Suppose is not continuous on (0,1). What can you say, if anything, about the extreme values off on (0,1? (e) Sketch the graph of a continuous function on (0,3) with a local minimum but no absolute minimum.
5. Suppose f : [0,1] → R is continuous, and in) is a Cauchy sequence in [0,1]. Prove or disprove: {f(In)} is a Cauchy sequence.
3. Suppose that f [0,1(0,1) is a non-decreasing function (NOT assumed to be continuous). Prove or disprove that there exists x E (0,1) such that f(x)-x
Suppose f is a continuous and differentiable function on [0,1] and f(0)= f(1). Let a E (0, 1). Suppose Vr,y(0,1) IF f'(x) 0 and f'(y) ±0 THEN f'(x) af'(y) Show that there is exactly f(ax) and f'(x) 0 such that f(x) one Hint: Suppose f(x) is a continuous function on [0, 1] and f(0) x € (0, 1) such that f(x) = f(ax) f(1). Let a e (0,1), there exists an
Suppose f is a continuous and differentiable function on...
real analysis
II. Consider the function f:[0,1] - R defined by f(x) 0 if x E [0,1]\ Q and f(x) = 1/q if x = p/q in lowest terms. 1. Prove that f is discontinuous at every x E Qn [0,1]. 2. Prove that f is continuous at every x e [0,1] \ Q.
II. Consider the function f:[0,1] - R defined by f(x) 0 if x E [0,1]\ Q and f(x) = 1/q if x = p/q in lowest...
14. If y =f(z) is a continuous function in a neighborhood around f'(c) = 0, does there have to be a local extrema on the graph of u = f(x) at x = c. 15. If/"(z) =-4(-7)2(z + 1) and the domain of f(x) is all real numbers, determine where f(x) is concave up, concave down and find any r-values of inflection points.
14. If y =f(z) is a continuous function in a neighborhood around f'(c) = 0, does there...
9. Is the function f(x) = sin 1/x continuous on (0,1)? Is it uniformly con- tinuous on (0,1). Justify your answers. 10. Is the function f(x) = x sin 1/x uniformly continuous on (0, 1)? Justify your answer.
Problem 5 Let f : [0,1] → R be continuous and assume f(zje (0, 1) for all x E (0,1). Let n E N with n 22. Show that there is eractly one solution in (0,1) for the equation 7L IC nx+f" (t) dt-n-f(t) dt.