For a metric space (X, d) and a continuous functionf from X into itself, show that the set of points {x: f(x) = x} is a closed subset of X. If f is a continuous function from a compact metric space (X, dx) into an arbitrary metric space (Y, dy), prove that f is uniformly continuous and the image of X under f is compact. Further, in addition if (X, dx) and (Y, dy) both are homeomorphic, what can be concluded about Y? Justify. Hence prove that there exist x1, x2 E [a, b] such that f(x1) = supxe[a,b}f (x) and f(x2) = infxe[a,b]f (x), where f: [a,b] → R is such that f(t) = at + B, for some a and ß in R. %3D

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