Consider g(x)=f(x)−x.
f(a)≥a so g(a)=f(a)−a≥0.
f(b)≤b so g(b)=f(b)−b≤0.
By the Intermediate Value Theorem, since g is continuous and 0∈[g(b),g(a)] there exists c∈[a,b] such that g(c)=f(c)−c=0, so f(c)=c for some c∈[a,b].
] → [a, 시 be continuous. Prove that there exists c E [a,b (3) Let f...
Suppose that f : [a, b] → a, b] is continuous. Prove that f has a fixed point, i.e., prove that there exists ce [a, b] such that f(c) = c.
Let f: [a, b] → [a,b] be a continuous function, where a, b are real numbers with a < b. Show that f has a fixed point (i.e., there exists x e [a, b] such that f(x) = x).
3. (a) Suppose f : (a, b) + R is differentiable, and there exists M E R such that If'(x) < M for all x € (a, b). Prove that f is uniformly continuous on (a, b). (b) Let f : [0, 1] → [0, 1] be a continuous function. Prove that there exists a point pe [0, 1] with f(p) = p.
3) Prove that there exists f : R → R non-negative and continuous such that f € L'OR : dm) ( i.e. SR \f|dm <00) and lim sup f(x) = ∞. 2-0
Suppose that f' exists and is continuous on a nonempty open interval (a,b) with f'(c) + 0 for all 2 € (a,b). | Prove that f is one-to-one on (a, b) and that f((a,b)) is an open interval II: if (c,d) is the open interval from (i), show that f-1EC'((c,d)), i.e. f-1 has a continuous first derivative on (a, b).
Let f be defined on an open interval I containing a point a (1) Prove that if f is differentiable on I and f"(a) exists, then lim h-+0 (a 2 h2 (2) Prove that if f is continuous at a and there exist constants α and β such that the limit L := lim h2 exists, then f(a)-α and f'(a)-β. Does f"(a) exist and equal to 2L?
Let f be defined on an open interval I containing a point a...
Given f, g E Da,b] , prove that min(f, g) E Da,시
6.59. Let f be a continuous function on [a, b]. Suppose that there exists a positive constant K such that If(x) <K for all x in [a, b]. Prove that f(x) = 0 for all x in [a, b]. *ſ isoidi,
Problem 1. Suppose that f:(a,b) + R is a continuous function and there exists a point p e (a, b) such that f' exists and is bounded on (a,b) {p}. Prove that f is uniformly continuous on (a,b).
Question 3. (4 marks) Let C([a, b]; R) be the space of all continuous functions on [a, b], 0 <a<b with the metric || f – 9|| = maxasaso \f (x) – g(x)]. For each f e C([a, b]; R), define a map F(f) by F(f)(x) = x5 + Vx € (a,b]. (65 – a5) Prove that there is a unique fixed point of F in the space C([a, b]; R); i.e. there is a unique fe C([a,b); R) such...