Give an example of a continuous function f : R → R that is diffierentiable everywhere except at 0 and 1
Give an example of a continuous function f : R → R that is diffierentiable everywhere except at 0 and 1
(1)Give an example of a function f : (0, 1) → R which is continuous, but such that there is no continuous function g : [0, 1] → R which agrees with f on (0, 1). (2)Suppose f : A (⊂ Rn) → R. Prove that if f is uniformly continuous then there is a unique continuous function g : B → R which agrees with f on A.(B is closure of A)
2. Find the value of c so that the function is continuous everywhere. f(x) = 02 – 22 r<2 1+c => 2 {
7. Give an example of a functionf 0,R that is discontinuous at every point of [0, 1 but such that |f is continuous on [O, 1] 7. Give an example of a functionf 0,R that is discontinuous at every point of [0, 1 but such that |f is continuous on [O, 1]
(8) Give an example of a metric space (X, d), a o-algebra A in X, and a continuous function f X ->R which is not measurable with respect to A. (Does this question make sense if (X, A) is a measurable space without a metric?) (8) Give an example of a metric space (X, d), a o-algebra A in X, and a continuous function f X ->R which is not measurable with respect to A. (Does this question make sense...
(1) If f: R₃ R a continuous function such that f(x)² > 0 for all xER. Show that either f(x) >0 for all a ER or f(x) <0 all X E R.
Question 2.1. . (i) Give an example of a function, f: R R, that is not bounded. (ii) Give an example of a function, f: (1.2) + R, that is not bounded. (iii) Give an example of a function, f: R → R. and a set. S. so that f attains its maximum on S. (iv) Give an example of a function, f: R R , and a set, S, so that f does not attain its maximum on S....
the function y=f(x)={ 0-4), 14x+16, x20 x<0 Consider 1. (a) Sketch the graph off. (3 pts.) (b) Verify that the function is continuous everywhere using the properties of the definition and possibly calculating the limit at a particular point. (2 pts.) (c) Show f'(x) is not continuous at x-0. (5 pts.) the function y=f(x)={ 0-4), 14x+16, x20 x
(i) Give an example of a function f(x,y) that is defined and continuous on the closed unit disk B(0) ((,y) E R2 but does not achieve a maximum on the punc- 2 marks] tured closed disk B.(0 )"-{ (z, y) E R2 10c x2 + y2 < 1} (i) Give an example of a function f(x,y) that is defined and continuous on the closed unit disk B(0) ((,y) E R2 but does not achieve a maximum on the punc- 2...
QUESTION 4 Find the intervals on which the function is continuous. у зе continuous everywhere discontinuous only when discontinuous only when e discontinuous only when e QUESTION 5 Provide an appropriate response. Use a calculator to graph the function f to see whether it appears to have a continuous extension to the origin. If it does, use Trace and Zoom to find a good candidate for the extended function's value at x 0. If the function does the origin from...
5. Consider the 'top-hat' function which is zero everywhere except between -1 and 1 where it takes value 1. Show, using only elementary integration, that the convolution of this function with itself gives a 'triangular function', 0 0 x〉2 which you should sketch Find the Fourier transform of the 'triangular function' f(x) using the result for the Fourier transform of a convolution. 5. Consider the 'top-hat' function which is zero everywhere except between -1 and 1 where it takes value...