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Suppose that g f is surjective Let z 2C Then since g f is surjective, there exists x 2A such that (g f)(x) = g(f(x)) = z Therefore if we let y = f(x) 2B, then g(y) = z Thus g is surjective Problem 338 In each part of the exercise, give examples of sets A;B;C and functions f A !B and g B !C satisfying the indicated properties (a) g.  · In mathematics, the mean value theorem states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints It is one of the most important results in real analysisThis theorem is used to prove statements about a function on an interval starting from local hypotheses about. ± & J G * r > Q y J 5 ö A &63 a X { g 4 k V r B a X { g 4 k Q y J ¹ µ » C &6& a X { g 4 k k q B a X { g 4 k Q?.

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