Suppose that f is uniformly continuous on(a,b). Let xn ∈ (a,b)converge to b as n→ ∞. Then{xn}is Cauchy;
hence, by Lemma 3.38, so is{f(xn)}. In particular, g(b) := lim
n→∞f(xn)
exists. This value does not change if we use a different sequence to a approximate b.
WEN-CHINGLIEN Advanced Calculus (I)
Proof:
Suppose that f is uniformly continuous on(a,b). Let xn ∈ (a,b)converge to b as n→ ∞. Then{xn}is Cauchy;
hence, by Lemma 3.38, so is{f(xn)}. In particular, g(b) := lim
n→∞f(xn)
exists. This value does not change if we use a different sequence to a approximate b.
WEN-CHINGLIEN Advanced Calculus (I)
Proof:
Suppose that f is uniformly continuous on(a,b). Let xn ∈ (a,b)converge to b as n→ ∞. Then{xn}is Cauchy;
hence, by Lemma 3.38, so is{f(xn)}. In particular, g(b) := lim
n→∞f(xn)
exists. This value does not change if we use a different sequence to a approximate b.
WEN-CHINGLIEN Advanced Calculus (I)
Proof:
Suppose that f is uniformly continuous on(a,b). Let xn ∈ (a,b)converge to b as n→ ∞. Then{xn}is Cauchy;
hence, by Lemma 3.38, so is{f(xn)}. In particular, g(b) := lim
n→∞f(xn)
exists. This value does not change if we use a different sequence to a approximate b.
WEN-CHINGLIEN Advanced Calculus (I)
Proof:
Suppose that f is uniformly continuous on(a,b). Let xn ∈ (a,b)converge to b as n→ ∞. Then{xn}is Cauchy;
hence, by Lemma 3.38, so is{f(xn)}. In particular, g(b) := lim
n→∞f(xn)
exists. This value does not change if we use a different sequence to a approximate b.
WEN-CHINGLIEN Advanced Calculus (I)
Proof:
Suppose that f is uniformly continuous on(a,b). Let xn ∈ (a,b)converge to b as n→ ∞. Then{xn}is Cauchy;
hence, by Lemma 3.38, so is{f(xn)}. In particular, g(b) := lim
n→∞f(xn)
exists. This value does not change if we use a different sequence to a approximate b.
WEN-CHINGLIEN Advanced Calculus (I)
Proof:
Suppose that f is uniformly continuous on(a,b). Let xn ∈ (a,b)converge to b as n→ ∞. Then{xn}is Cauchy;
hence, by Lemma 3.38, so is{f(xn)}. In particular, g(b) := lim
n→∞f(xn)
exists. This value does not change if we use a different sequence to a approximate b.
WEN-CHINGLIEN Advanced Calculus (I)
Indeed, let yn ∈ (a,b)be another sequence that converges to b as n→ ∞. Givenǫ >0, chooseδ >0 such that (9) holds for E = (a,b). Since xn−yn →0, choose N ∈N so that n ≥N implies|xn−yn| < δ. By (9), then,|f(xn) −f(yn)| < ǫfor all n ≥N. Taking the limit of this inequality as n → ∞, we obtain
| lim
n→∞f(xn) − lim
n→∞f(yn)| ≤ ǫ
for all ǫ >0. It follows from Theorem 1.9 that
n→∞lim f(x)= lim
n→∞f(yn).
WEN-CHINGLIEN Advanced Calculus (I)
Indeed,let yn ∈ (a,b)be another sequence that converges to b as n→ ∞. Given ǫ >0,chooseδ >0 such that (9) holds for E = (a,b). Since xn−yn →0, choose N ∈N so that n ≥N implies|xn−yn| < δ. By (9), then,|f(xn) −f(yn)| < ǫfor all n ≥N. Taking the limit of this inequality as n → ∞, we obtain
| lim
n→∞f(xn) − lim
n→∞f(yn)| ≤ ǫ
for all ǫ >0. It follows from Theorem 1.9 that
n→∞lim f(x)= lim
n→∞f(yn).
WEN-CHINGLIEN Advanced Calculus (I)
Indeed, let yn ∈ (a,b)be another sequence that converges to b as n→ ∞. Givenǫ >0, chooseδ >0 such that (9) holds for E = (a,b). Since xn−yn →0, choose N ∈N so that n ≥N implies|xn−yn| < δ. By (9), then,|f(xn) −f(yn)| < ǫfor all n ≥N. Taking the limit of this inequality as n → ∞, we obtain
| lim
n→∞f(xn) − lim
n→∞f(yn)| ≤ ǫ
for all ǫ >0. It follows from Theorem 1.9 that
n→∞lim f(x)= lim
n→∞f(yn).
WEN-CHINGLIEN Advanced Calculus (I)
Indeed, let yn ∈ (a,b)be another sequence that converges to b as n→ ∞. Given ǫ >0,chooseδ >0 such that (9) holds for E = (a,b). Since xn−yn →0, choose N ∈N so that n ≥N implies|xn−yn| < δ. By (9), then,|f(xn) −f(yn)| < ǫfor all n ≥N. Taking the limit of this inequality as n → ∞, we obtain
| lim
n→∞f(xn) − lim
n→∞f(yn)| ≤ ǫ
for all ǫ >0. It follows from Theorem 1.9 that
n→∞lim f(x)= lim
n→∞f(yn).
WEN-CHINGLIEN Advanced Calculus (I)
Indeed, let yn ∈ (a,b)be another sequence that converges to b as n→ ∞. Given ǫ >0, chooseδ >0 such that (9) holds for E = (a,b). Since xn−yn →0, choose N ∈N so that n ≥N implies|xn−yn| < δ. By (9), then,|f(xn) −f(yn)| < ǫfor all n ≥N. Taking the limit of this inequality as n → ∞, we obtain
| lim
n→∞f(xn) − lim
n→∞f(yn)| ≤ ǫ
for all ǫ >0. It follows from Theorem 1.9 that
n→∞lim f(x)= lim
n→∞f(yn).
WEN-CHINGLIEN Advanced Calculus (I)
Indeed, let yn ∈ (a,b)be another sequence that converges to b as n→ ∞. Given ǫ >0, chooseδ >0 such that (9) holds for E = (a,b). Since xn−yn →0, choose N ∈N so that n ≥N implies|xn−yn| < δ. By (9), then,|f(xn) −f(yn)| < ǫfor all n ≥N. Taking the limit of this inequality as n → ∞, we obtain
| lim
n→∞f(xn) − lim
n→∞f(yn)| ≤ ǫ
for all ǫ >0. It follows from Theorem 1.9 that
n→∞lim f(x)= lim
n→∞f(yn).
WEN-CHINGLIEN Advanced Calculus (I)
Indeed, let yn ∈ (a,b)be another sequence that converges to b as n→ ∞. Given ǫ >0, chooseδ >0 such that (9) holds for E = (a,b). Since xn−yn →0, choose N ∈N so that n ≥N implies|xn−yn| < δ. By (9), then,|f(xn) −f(yn)| < ǫfor all n ≥N. Taking the limit of this inequality as n → ∞,we obtain
| lim
n→∞f(xn) − lim
n→∞f(yn)| ≤ ǫ
for all ǫ >0. It follows from Theorem 1.9 that
n→∞lim f(x)= lim
n→∞f(yn).
WEN-CHINGLIEN Advanced Calculus (I)
Indeed, let yn ∈ (a,b)be another sequence that converges to b as n→ ∞. Given ǫ >0, chooseδ >0 such that (9) holds for E = (a,b). Since xn−yn →0, choose N ∈N so that n ≥N implies|xn−yn| < δ. By (9), then,|f(xn) −f(yn)| < ǫfor all n ≥N. Taking the limit of this inequality as n → ∞, we obtain
| lim
n→∞f(xn) − lim
n→∞f(yn)| ≤ ǫ
for all ǫ >0. It follows from Theorem 1.9 that
n→∞lim f(x)= lim
n→∞f(yn).
WEN-CHINGLIEN Advanced Calculus (I)
Indeed, let yn ∈ (a,b)be another sequence that converges to b as n→ ∞. Given ǫ >0, chooseδ >0 such that (9) holds for E = (a,b). Since xn−yn →0, choose N ∈N so that n ≥N implies|xn−yn| < δ. By (9), then,|f(xn) −f(yn)| < ǫfor all n ≥N. Taking the limit of this inequality as n → ∞,we obtain
| lim
n→∞f(xn) − lim
n→∞f(yn)| ≤ ǫ
for all ǫ >0. It follows from Theorem 1.9 that
n→∞lim f(x)= lim
n→∞f(yn).
WEN-CHINGLIEN Advanced Calculus (I)
Indeed, let yn ∈ (a,b)be another sequence that converges to b as n→ ∞. Given ǫ >0, chooseδ >0 such that (9) holds for E = (a,b). Since xn−yn →0, choose N ∈N so that n ≥N implies|xn−yn| < δ. By (9), then,|f(xn) −f(yn)| < ǫfor all n ≥N. Taking the limit of this inequality as n → ∞, we obtain
| lim
n→∞f(xn) − lim
n→∞f(yn)| ≤ ǫ
for all ǫ >0. It follows from Theorem 1.9 that
n→∞lim f(x)= lim
n→∞f(yn).
WEN-CHINGLIEN Advanced Calculus (I)
Indeed, let yn ∈ (a,b)be another sequence that converges to b as n→ ∞. Given ǫ >0, chooseδ >0 such that (9) holds for E = (a,b). Since xn−yn →0, choose N ∈N so that n ≥N implies|xn−yn| < δ. By (9), then,|f(xn) −f(yn)| < ǫfor all n ≥N. Taking the limit of this inequality as n → ∞, we obtain
| lim
n→∞f(xn) − lim
n→∞f(yn)| ≤ ǫ
for all ǫ >0. It follows from Theorem 1.9 that
n→∞lim f(x)= lim
n→∞f(yn).
WEN-CHINGLIEN Advanced Calculus (I)
Thus, g(b)is well defined. A similar argument defines g(a). Set g(x) =f(x)for x ∈ (a,b). Then g is defined on [a,b], satifies (11), and is continuous on[a,b]by the Sequential Characterization of Limits. Thus, f can be
“continuously extended” to g as required.
Conversely, suppose that there is a function g
continuouos on [a,b] that satifies (11). By theorem 3.39, g is uniformly continuous on [a,b]; hence, g is uniformly continuous on (a,b). We conclude that f is uniformly continuous on (a,b).2
WEN-CHINGLIEN Advanced Calculus (I)
Thus, g(b)is well defined. A similar argument defines g(a). Set g(x) =f(x)for x ∈ (a,b). Then g is defined on [a,b], satifies (11), and is continuous on[a,b]by the Sequential Characterization of Limits. Thus, f can be
“continuously extended” to g as required.
Conversely, suppose that there is a function g
continuouos on [a,b] that satifies (11). By theorem 3.39, g is uniformly continuous on [a,b]; hence, g is uniformly continuous on (a,b). We conclude that f is uniformly continuous on (a,b).2
WEN-CHINGLIEN Advanced Calculus (I)
Thus, g(b)is well defined. A similar argument defines g(a). Set g(x) =f(x)for x ∈ (a,b). Then g is defined on [a,b], satifies (11), and is continuous on[a,b]by the Sequential Characterization of Limits. Thus,f can be
“continuously extended” to g as required.
Conversely, suppose that there is a function g
continuouos on [a,b] that satifies (11). By theorem 3.39, g is uniformly continuous on [a,b]; hence, g is uniformly continuous on (a,b). We conclude that f is uniformly continuous on (a,b).2
WEN-CHINGLIEN Advanced Calculus (I)
Thus, g(b)is well defined. A similar argument defines g(a). Set g(x) =f(x)for x ∈ (a,b). Then g is defined on [a,b], satifies (11), and is continuous on[a,b]by the Sequential Characterization of Limits. Thus, f can be
“continuously extended” to g as required.
Conversely, suppose that there is a function g
continuouos on [a,b] that satifies (11). By theorem 3.39, g is uniformly continuous on [a,b]; hence, g is uniformly continuous on (a,b). We conclude that f is uniformly continuous on (a,b).2
WEN-CHINGLIEN Advanced Calculus (I)
Thus, g(b)is well defined. A similar argument defines g(a). Set g(x) =f(x)for x ∈ (a,b). Then g is defined on [a,b], satifies (11), and is continuous on[a,b]by the Sequential Characterization of Limits. Thus,f can be
“continuously extended” to g as required.
Conversely, suppose that there is a function g
continuouos on [a,b] that satifies (11). By theorem 3.39, g is uniformly continuous on [a,b]; hence, g is uniformly continuous on (a,b). We conclude that f is uniformly continuous on (a,b).2
WEN-CHINGLIEN Advanced Calculus (I)
Thus, g(b)is well defined. A similar argument defines g(a). Set g(x) =f(x)for x ∈ (a,b). Then g is defined on [a,b], satifies (11), and is continuous on[a,b]by the Sequential Characterization of Limits. Thus, f can be
“continuously extended” to g as required.
Conversely, suppose that there is a function g
continuouos on [a,b] that satifies (11). By theorem 3.39, g is uniformly continuous on [a,b]; hence, g is uniformly continuous on (a,b). We conclude that f is uniformly continuous on (a,b).2
WEN-CHINGLIEN Advanced Calculus (I)
Thus, g(b)is well defined. A similar argument defines g(a). Set g(x) =f(x)for x ∈ (a,b). Then g is defined on [a,b], satifies (11), and is continuous on[a,b]by the Sequential Characterization of Limits. Thus, f can be
“continuously extended” to g as required.
Conversely, suppose that there is a function g
continuouos on [a,b] that satifies (11). By theorem 3.39, g is uniformly continuous on [a,b]; hence, g is uniformly continuous on (a,b). We conclude that f is uniformly continuous on (a,b).2
WEN-CHINGLIEN Advanced Calculus (I)
Thus, g(b)is well defined. A similar argument defines g(a). Set g(x) =f(x)for x ∈ (a,b). Then g is defined on [a,b], satifies (11), and is continuous on[a,b]by the Sequential Characterization of Limits. Thus, f can be
“continuously extended” to g as required.
Conversely, suppose that there is a function g
continuouos on [a,b] that satifies (11). By theorem 3.39, g is uniformly continuous on [a,b];hence, g is uniformly continuous on (a,b). We conclude that f is uniformly continuous on (a,b).2
WEN-CHINGLIEN Advanced Calculus (I)
Thus, g(b)is well defined. A similar argument defines g(a). Set g(x) =f(x)for x ∈ (a,b). Then g is defined on [a,b], satifies (11), and is continuous on[a,b]by the Sequential Characterization of Limits. Thus, f can be
“continuously extended” to g as required.
Conversely, suppose that there is a function g
continuouos on [a,b] that satifies (11). By theorem 3.39, g is uniformly continuous on [a,b]; hence, g is uniformly continuous on (a,b). We conclude that f is uniformly continuous on (a,b).2
WEN-CHINGLIEN Advanced Calculus (I)
Thus, g(b)is well defined. A similar argument defines g(a). Set g(x) =f(x)for x ∈ (a,b). Then g is defined on [a,b], satifies (11), and is continuous on[a,b]by the Sequential Characterization of Limits. Thus, f can be
“continuously extended” to g as required.
Conversely, suppose that there is a function g
continuouos on [a,b] that satifies (11). By theorem 3.39, g is uniformly continuous on [a,b];hence, g is uniformly continuous on (a,b). We conclude that f is uniformly continuous on (a,b).2
WEN-CHINGLIEN Advanced Calculus (I)
Thus, g(b)is well defined. A similar argument defines g(a). Set g(x) =f(x)for x ∈ (a,b). Then g is defined on [a,b], satifies (11), and is continuous on[a,b]by the Sequential Characterization of Limits. Thus, f can be
“continuously extended” to g as required.
Conversely, suppose that there is a function g
continuouos on [a,b] that satifies (11). By theorem 3.39, g is uniformly continuous on [a,b]; hence, g is uniformly continuous on (a,b). We conclude that f is uniformly continuous on (a,b).2
WEN-CHINGLIEN Advanced Calculus (I)
Thus, g(b)is well defined. A similar argument defines g(a). Set g(x) =f(x)for x ∈ (a,b). Then g is defined on [a,b], satifies (11), and is continuous on[a,b]by the Sequential Characterization of Limits. Thus, f can be
“continuously extended” to g as required.
Conversely, suppose that there is a function g
continuouos on [a,b] that satifies (11). By theorem 3.39, g is uniformly continuous on [a,b]; hence, g is uniformly continuous on (a,b). We conclude that f is uniformly continuous on (a,b).2
WEN-CHINGLIEN Advanced Calculus (I)
Thus, g(b)is well defined. A similar argument defines g(a). Set g(x) =f(x)for x ∈ (a,b). Then g is defined on [a,b], satifies (11), and is continuous on[a,b]by the Sequential Characterization of Limits. Thus, f can be
“continuously extended” to g as required.
Conversely, suppose that there is a function g
continuouos on [a,b] that satifies (11). By theorem 3.39, g is uniformly continuous on [a,b]; hence, g is uniformly continuous on (a,b). We conclude that f is uniformly continuous on (a,b).2
WEN-CHINGLIEN Advanced Calculus (I)
Example:
Prove that f(x) = (x −1)
log x is uniformly continuous on (0,1).
WEN-CHINGLIEN Advanced Calculus (I)
Example:
Prove that f(x) = (x −1)
log x is uniformly continuous on (0,1).
WEN-CHINGLIEN Advanced Calculus (I)