Let f 0 1 leftarrow Q be a continuous function such that f0

Let f: [0, 1] leftarrow Q be a continuous function such that f(0) = 0. Construct an argument to determine the value of f(squareroot2/2). Let f be a continuous function from R into R^2. Construct an argument to determine whether the set {x ELEMENT OF WITH VERTICAL BAR AT END OF HORIZONTAL STROKE R |f(x) = (1, 1)} is closed in R Construct an argument to determine whether the set F = {(x, y) ELEMENT OF WITH VERTICAL BAR AT END OF HORIZONTAL STROKE R^2 | x^2 +y^2 lesserthanequalto 9} N-ARY UNION {(x, 0) ELEMENT OF WITH VERTICAL BAR AT END OF HORIZONTAL STROKE R^2 | 3

Solution

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we ae given that the domain for f is [o,1] and the range for f is Q . Q represents the set of all rational numbers

now we are given that f(0) = 0

and we know a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, p and q, with the denominator q not equal to zero.

now sqrt(2)/2 lies within the domain [0 , 1]

but f(sqrt(2)/2) will not be equal to sqrt(2)/2 as sqrt(2) is not a rational number

so f(sqrt(2)/2) need to be some rational numebr

so f(sqrt(2)/2) = Q

and example of such a function wich is continuous and give only rational values as the range is

f(x) = x^2 , here for any x value between [0,1] we\'ll get a rational number Q as the f value