1. Consider a set X equipped with the distance d defined by
d(x.y) = 0 if x=y
and a metric space Y. Determine all continuous maps from x to Y.
2. Let (X,d) be a metric space and a € X. Assume that u, U and f are real-valued
functions on X such that
for all . € x and that
lim u(x) = e S " = lim v(x).
Prove that for any E > 0, there exists r >0 such that
for any x € B(a,r).
3. The indicator function of Q is defined on R by
Determine all the points where Xg is continuous.
4. Let f IR - R be a function that is continuous at 0 and additive, that is,
for every x. y, € IR,
(a) Prove that f(nx) = /(x) for any n €Z and any x € R.
(b) Prove that f(r) = f(1) for any r € Q.
(c) Prove that f is continuous on R and derive an expression of f(x) for any I € R.
5. The function E is defined on IR by E(x) = where
is the only integer n € Z
72 SoThe purpose of this problem is to study the function defined on IR by
(a) Calculate f(x) for r < - 1 and for x > 1.
(b) Determine f(x) for I € (P+ and verify that f is continuous on
(c) Study the continuity of f on the right of
(d) Is f continuous at 0?
Hint: prove that -
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