###### Answer: B. (-∞ 2)

Thus the function g(x) = f(x) − x is a continuous real valued function which is positive at x = 0 and negative at x = 1 By the intermediate value theorem there is some point x 0 with g(x 0) = 0 which is to say that f(x 0) − x 0 = 0 and so x 0 is a fixed point. The open interval does not have the fixed-point property. The mapping f(x) = x 2 has no fixed point on the interval (0 1). The closed disc

If g is of bounded variation and f is bounded then it is possible to write = − where g 1 ( x ) = V x a g is the total variation of g in the interval [a x ] and g 2 ( x ) = g 1 ( x ) − g ( x ). Both g 1 and g …

This captures an intuitive property of continuous functions over the real numbers: given f continuous on [1 2] with the known values f (1) = 3 and f (2) = 5 then the graph of y = f ( x ) must pass through the horizontal line y = 4 while x moves from 1 to 2. It represents the idea that the graph of a continuous function on a closed interval can be drawn without lifting a pencil from the paper.

This means that the upper and lower sums of the function f are evaluated on a partition a = x 0 ≤ x 1 ≤ . . . ≤ x n = b whose values x i are increasing. Geometrically this signifies that integration takes place “left to right” evaluating f within intervals [ x i x i +1 ] where an interval …

The σ-algebra of Borel sets is an important structure on real numbers. If X has its σ-algebra and a function f is such that the preimage f −1 (B) of any Borel set B belongs to tha…