 # PROBLEM # 1 Let f : [a, [c, d] be a continuous function. Suppose ...

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PROBLEM # 1 Let f : [a, [c, d] be a continuous function. Suppose that g : [a, b] R is another continuous function such that there exists x1, x2 € [a, b] with the property g(x1) = c and g (x2) = d. Prove that there exists some xo € [a, b] with f(zo)=g(zo). = PROBLEM # 2 Let f : [a, b] a, b] be a continuous function with If(x) - f(y) < x - y for all x # y € [a,b]. Prove that there exists a unique xo € [a, b] with f (xo) = xo, and moreover that the iterated composition fn = fofo.. f (n times) is such that xn = fn '(x) xo for all x € a, b]. PROBLEM # 3 Consider the following unctionf:R-R: 1 if x=0; f(x) = n 1/2 , if x = m n where m E Z , n € N and gcd(n,m) = 1; 0 , if Show that f is continuous at all x € R)Q, and discontinuous at all x € Q. PROBLEM # 4 Let p(x) : R R be a monic odd degree polynomial, i.e. p(x) = Er-sakach where d is odd and ad = 1. Suppose there exists X1 < x2 with p(x2) <0Show that the polynomial p(x) = x3 + 4x² + x - 1 factors with all real roots. PROBLEM #5 Let f : D R, and let xo € D. We say "f is left continuous at x0" if whenever In € D and Xn < xo, then xn xo implies f (xn) f (xx). Likewise we say "f is right continuous at xo" if whenever xn € D and xn > xo, then xn xo implies f (xn) f(xo). Show that a function f : D R is continuous at xo € D iff it is both right and left continuous as xo. PROBLEM #6 A function f : D R is called lower semicontinuous at xo € D if f < lim inf f(xn) for all sequences xn € D with Xn xo € D. A function f : D R is called upper semicontinuous at xo € D if lim sup f(xn) < f(xo) for all sequences xn € D with xn xo € D. a) Prove that a function f : D R is continuous at xo € D iff it is both upper and lower semicontinuous at xo. b) Prove that a monotone increasing left continuous function (see Problem #5 above) is always lower semi- continuous. c) Prove that a monotone increasing right continuous function (see Problem #5 above) is always upper semicontinuous. d) Prove that the function from Problem #3 above is upper semicontinuous at all points.

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