Give an example of a function f which satisfies the intermediate-value property on a closed and bounded interval [a, b] but is not continuous on a, 6). () Give an example of a function f which is monotone increasing on a nd bounded interval (a, b] but does not satisfy the intermediate-value property on (a, b). 2. Let ceR and a function f : R -R is continuous at c. If for every positive S there is a point y in (e-6, c+ 6) such that f(y) = 0, prove that f(c) = 0. 3. Let f:R- R be continuous on R and let c e R such that f(c) >p. Prove that there exists a neighbourhood U of c such that f(r) >p for all zEU. 4. Let a function f: R - R be continuous on R. Prove that the set 2(f) = (zER: f(2) = 0} is a closed set in R. Give an example of a function f continuous on R such that 1) Z) is a bounded enumerable set; (ii) Z(f) is an unbounded enumerable %3D set. 5. Let f: R-R be continuous on R. A point ce R is said to be a fized point of f if f(c) = c holds. Prove that the set of all fixed points of f is a closed set. 6. Let I = la, b) be a closed and bounded interval and a function f:1-R be Continuous on I and f(z) >0 for all z E I. Prove that there exists a positive number a such that f(r) 2 a for all r I. . A function f: (0, 1] -- R in continuous on (0, 1) and f assumes only rational values on (0, 1]. Prove that f is a constant. 8. Let : [a,b) f(a) < g(a), f(b) > g(b). Show that there exists a point c in (a, b) such that f(e) = g(c). R and g: (a, b) - R be continuous on (a, b] and let 1 Deduce that cns z= for some r (0, 5).