Suppose f : [a,b] → R is a continuous function. Prove that the direct image f([a,b]) is a closed and bounded interval or a single number.

1. Consider the function fƒ: R → R defined by f(x) = = 2x + 1 5- 3x when x 1. A. Consider function f at the point x = 1. Is function f continuous at this point? Why or why not? B. Consider function f at a point x = C. Consider function f at a point x = a where a is less than 1. Is function f continuous at this point? Why or why not? a where a is greater than 1. Is function f continuous at this point? Why or why not?

Let ℓ be a line and let f : ℓ → R be a function such that PQ = |f(P) - f(Q)| for every P, Q ∈ ℓ. Prove that f is a coordinate function for ℓ.