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15. Suppose that X e Df CR" and X is a limit point of D f. Show that f is continuous at X if and only if limk→∞ in Df such that limk→. Xk = X. HINT: See the proof of Theorem 4.2.6. f (Xg) f(X) whenever {Xx} is a sequence of points

15. Suppose that X e Df CR" and X is a limit point of D f. Show that f is continuous at X if and only if limk→∞ in Df such that limk→. Xk = X. HINT: See the proof of Theorem 4.2.6. f (Xg) f(X) whenever {Xx} is a sequence of points

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Theorem 4.2.6 Let f be defined on a closed interval [a, b] containing x. Then f is continuous at x (from the right if x = a, from the left if x = b) if and only if lim f(xn) = f(x) (4.2.6) whenever {xn} is a sequence of points in [a, b] such that lim xn = x. (4.2.7) Proof Assume that a < x < b; only minor changes in the proof are needed if x = a or X = b. First, suppose that f is continuous at x and {xn} is a sequence of points in [a, b] satisfying (4.2.7). If e > 0, there is a 8 > 0 such that |f (x) – f(X)| <e if |x – x| < 8. (4.2.8) – x| < 8 if n > N. This and (4.2.8) From (4.2.7), there is an integer N such that |Xn imply that | f (xn) – f (x)| < e if n > N. This implies (4.2.6), which shows that the stated condition is necessary. For sufficiency, suppose that f is discontinuous at x. Then there is an eo > 0 such that, for each positive integer n, there is a point x, that satisfies the inequality 1 |Xn – x| <
Transcribed Image Text:Theorem 4.2.6 Let f be defined on a closed interval [a, b] containing x. Then f is continuous at x (from the right if x = a, from the left if x = b) if and only if lim f(xn) = f(x) (4.2.6) whenever {xn} is a sequence of points in [a, b] such that lim xn = x. (4.2.7) Proof Assume that a < x < b; only minor changes in the proof are needed if x = a or X = b. First, suppose that f is continuous at x and {xn} is a sequence of points in [a, b] satisfying (4.2.7). If e > 0, there is a 8 > 0 such that |f (x) – f(X)| <e if |x – x| < 8. (4.2.8) – x| < 8 if n > N. This and (4.2.8) From (4.2.7), there is an integer N such that |Xn imply that | f (xn) – f (x)| < e if n > N. This implies (4.2.6), which shows that the stated condition is necessary. For sufficiency, suppose that f is discontinuous at x. Then there is an eo > 0 such that, for each positive integer n, there is a point x, that satisfies the inequality 1 |Xn – x| <
Suppose that X e Df C R" and X is a limit point of D f. Show that f is continuous at X if and only if lim→∞ f(Xk) = in Df such that limk→ Xk = X. HINT: See the proof of Theorem 4.2.6. 15. f (X) whenever {Xk} is a sequence of points
Transcribed Image Text:Suppose that X e Df C R" and X is a limit point of D f. Show that f is continuous at X if and only if lim→∞ f(Xk) = in Df such that limk→ Xk = X. HINT: See the proof of Theorem 4.2.6. 15. f (X) whenever {Xk} is a sequence of points
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