We will use the following lemma to prove this ,
Lemma : Continue image of a compact set is compact that is if be a continuous function then is compact .
Given is continuous and positive function on then , .
Suppose there does not exist any such that the for all there exist such that .
Choose so for all there exist such that .
So for the above choose sequence the sequence converges to 0 in .
Now since there exist a sequence converges to 0 in so 0 a limit point of the set and since is compact so it contains all its limit points and so .
there exist such that , a contradiction to .
Hence there exist >0 such that for any
Exercise 5. Prove that if f is a continuous and positive function on (0,1], there exists...
Let f: [0,1]→R be uniformly continuous, so that for every >0, there exists δ >0 such that |x−y|< δ=⇒|f(x)−f(y)|< for every x,y∈[0,1].The graph of f is the set G f={(x,f(x)) :x∈[0,1]}.Show that G f has measure zero Let f : [0, 1] → R be uniformly continuous, so that for every e > 0, there exists 8 >0 such that 2- y<83|f() - f(y)< € for every 1, 9 € [0,1]. The graph of f is the set Gj =...
Let f : [0,1] → R be uniformly continuous, so that for every e > 0, there exists 8 >0 such that 2 - y<== f() - f(y)< € for every 2, Y € [0,1]. The graph of f is the set Gj = {, f(c)): 1 € [0,1]}. : Show that G has measure zero.
Let f : [0,1] → R be uniformly continuous, so that for every e > 0, there exists 8 >0 such that 2 - y<== f() - f(y)< € for every 2, Y € [0,1]. The graph of f is the set Gj = {, f(c)): 1 € [0,1]}. : Show that G has measure zero.
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,
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
= (a) Suppose that limx+c f(x) L > 0. Prove that there exists a 8 >0 such that if 0 < \x – c < 8, then f(x) > 0. (b) Use Part (a) and the Heine-Borel Theorem to prove that if is continuous on [a, b] and f(x) > 0 for all x € [a,b], then there exists an e > 0 such that f(x) > e for all x E [a, b].
Let U be an open subset of R". Let f: UCR" ->Rm. (a) Prove that f is continuously differentiable if and only if for each a e U, for eache > 0, there exists o > 0 such that for each xe U, if ||x - a| << ô, then |Df (x) Df(a)| < e.
Prove: By taking the following problem as being given/true : (Analysis on Metric Spaces) Let f : [0, 1] x [0, 1] + R be defined by f(x,y) = ſi if y=x? if y #r? Show that f is integrable on [0, 1] x [0,1]. Let f : [0, 1] + R be uniformly continuous, so that for every e > 0, there exists 8 >0 such that -y<= f(x) - f(y)< € for every I, Y E (0,1). The...
29.9) Compute the Fourier transform of the periodic function f(t) to prove the equation shown below: 29.9. Let f(t) = iftl > T. 0 Show that f(w) =
4. (a) Suppose that limz-c f(x) = L > 0. Prove that there exists a 8 >0 such that if 0 < 12 – c < 8, then f(x) > 0. (b) Use Part (a) and the Heine-Borel Theorem to prove that if is continuous on (a, b) and f(x) > 0 for all x € (a, b), then there exists an e > 0 such that f(x) > € for all x € [a, b].