construct a bijective function of f:[0, inf) -> R

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Expert Solution

We are going to define a bijective function from to . At first we define . Then we define the function interval-wise.

for all and . Observe that image of (0,1] under f is (0,1] and image of (1,2] is [-1,0). Similarly, we define , and . Again observe that image of (2,3] is (1,2] and image of (3,4] is [-2,-1).

Again we define and . Then the images will be (2,3] and [-3,-2) respectively. In general, we define and . Now it can be easily checked that is a bijection. Infact, injectivity of this function is trivial and for the surjectivity, let . If , then . If y>0, then for some . Then . If y<0, then for some n. Then . Hence is surjective.

Colby Messinger answered 1 year ago

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