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
Chapter3: Functions
Section3.3: Rates Of Change And Behavior Of Graphs
Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...
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Proof the following, I have provided the given hint in the images
![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| <](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F187d6ce3-f16e-4919-9795-1c801fb3fce4%2F56342ab5-388a-477f-bba4-d717b42a3276%2F3rbc0tm_processed.jpeg&w=3840&q=75)
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| <
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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