(Continuous Limit Theorem). Let (fn) be a sequence of functions defined on A ⊆ R that converges uniformly on A to a function f. If each fn is continuous at c ∈ A, then f is continuous at c. fm,n>Nf+eff-eAFigure 6.4: fn +f UNIFORMLY ON A.94 93 9291g+e9-€AFigure 6.5: In + g POINTWISE, BUT NOT UNIFORMLY.Proof. Fix c E A and let e > 0. Choose N so that Proof. Fix ce A and let e> 0. Choose N so that\fx (x) – f(x)| <for all a € A. Because fN is continuous, there exists a d > 0 for which\fN (x) – fN(c)| <:3is true whenever |x – c| < 8. But this implies\f (x) – f(c)||f(x) – fN (1) + fN (2) – fN (c) + fN (c) – f(c)|< If(x) – fN(x)| + \fN(x) – fN(c)| + \fv (c) – f(c)|+3+3= €.Thus, f is continuous at ce A.

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Description: (Continuous Limit Theorem). Let (fn) be a sequence of functions defined on A ⊆ R that converges uniformly on A to a function f. If each fn is continuous at c ∈ A, then f is continuous at c. fm,n>Nf+eff-eAFigure 6.4: fn +f UNIFORMLY ON A.94 93 9291g+e

Answered: fm,n>N f+e f f-e A Figure 6.4: fn +f…

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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(Continuous Limit Theorem). Let (fn) be a sequence of functions defined on A ⊆ R that converges uniformly on A to a function f. If each fn is continuous at c ∈ A, then f is continuous at c.

fm,n>N
f+e
f
f-e
A
Figure 6.4: fn +f UNIFORMLY ON A.
94 93 92
91
g+e
9-€
A
Figure 6.5: In + g POINTWISE, BUT NOT UNIFORMLY.
Proof. Fix c E A and let e > 0. Choose N so that

Proof. Fix ce A and let e> 0. Choose N so that
\fx (x) – f(x)| <
for all a € A. Because fN is continuous, there exists a d > 0 for which
\fN (x) – fN(c)| <:
3
is true whenever |x – c| < 8. But this implies
\f (x) – f(c)|
|f(x) – fN (1) + fN (2) – fN (c) + fN (c) – f(c)|
< If(x) – fN(x)| + \fN(x) – fN(c)| + \fv (c) – f(c)|
+
3
+
3
= €.
Thus, f is continuous at ce A.

Expert Solution

Step 1

Given: $f_{n}$ is a sequence of functions defined on $A \subseteq R$ that converges uniformly on $A$ to a function $f .$ Each $f_{n}$ is continuous at $c \in A .$
To prove: $f$ is continuous at $c \in A .$

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