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7. Give an example of a functionf 0,R that is discontinuous at every point of [0, 1 but such that...
1. (a) Give an example of functions f and g defined on R such that both f and g are discontinuous at every c, but the sum +9 is continuous at every (b) Give an example of functions and g defined on R such that g is not constant, / is discontinuous at everyx, but the composition gol is continuous at every
(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)
Give an example of a continuous function f : R → R that is diffierentiable everywhere except at 0 and 1
this is Topology 3) Ifa functionf(R,T.)-(R,T) įs continuous, then f(R,Ts)-(R, т)is continuous. 4) If a function EIR, ті )-(R,%) is continuous, then e (R, T)-cR,n) is continuous. 5) If a function f: (R )-(R, Tİ) İS continuous, then f(R,7, ) → (RM) t8 continuous. 6) If a function f:(R雨)-(RM) is continuous, then f (RM )-(RM) is continuous. 7) Any two discrete topological spaces are homeomorphic. 8) Any one-to-one, onto function between two discrete topological spaces is a homeomorphism
7 of 15 (0 complete) Find all values xa where the function is discontinuous. For each value of x, give the limit of the function as x approaches a. Be sure to note when the limit doesn't exist 7 + x f(x)= O Select the choice below and, if necessary, fill in the answer box/es) within your choice (Use a comma to separate answers as needed.) O A The function is discontinuous over the interval The limit is (Type your...
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...
1) In each case give an example of A C R and f : R → R continuous such that i A compact with (ii) A connected with f (A) no connected. ii A open with f(A) not open (iv) A closed with f(A) not closed A) no compact
PROJECT 6.2 In this project you will construct an increasing function that is discontinuous at each rational point in (0, 1) and continuous at each irrational point in (0, 1). We will need two basic facts: a. The rational numbers in the interval (0, 1) can be arranged in a sequence rThis is true because the set of rational numbers is countable. (See Example 0.12 and Corollary 0.15.) b. Any rearrangement of an absolutely convergent series converges, and any sub-...
The deflection along a discontinuous cantilever beam of length 4 units is governed y (0)-y' (0) 0 d2y dx2J 4) (4) (a) Show that 1+2H (-2)2) if o < <4 (b) Ealute dr dl e) Evaluate the deflection y(r). the deflection y (r Hint: If F (x) is an antiderivative of f (x) then f (x) H (r-a) dr = F (x)-F (a)] H (z-a) + C. The deflection along a discontinuous cantilever beam of length 4 units is governed...
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...