Are the following statements true or false? If true give a proof, and if false give a counter-example: (a) Assume that that a sequence of continuous functions fn converges uniformly to a function f on the interval (a, b). This means that: ∀ε>0 ∃N>0 ∀n>N ∀x∈[a,b] : |fn(x)−f(x)|<ε. Then the function f is also continuous. (b)Assume that that a sequence of differentiable functions fn converges uniformly to a function f on the interval (a, b). Then the function f is also differentiable
Are the following statements true or false? If true give a proof, and if false give a counter-example: (a) Assume that that a sequence of continuous functions fn converges uniformly to a function f on the interval (a, b). This means that: ∀ε>0 ∃N>0 ∀n>N ∀x∈[a,b] : |fn(x)−f(x)|<ε. Then the function f is also continuous. (b)Assume that that a sequence of differentiable functions fn converges uniformly to a function f on the interval (a, b). Then the function f is also differentiable
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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Are the following statements true or false? If true give a proof, and if false give a counter-example:
(a) Assume that that a sequence of continuous functions fn converges uniformly to a function f on the interval (a, b). This means that:
∀ε>0 ∃N>0 ∀n>N ∀x∈[a,b] : |fn(x)−f(x)|<ε.
Then the function f is also continuous.
(b)Assume that that a sequence of differentiable functions fn converges uniformly to a function f on the interval (a, b).
Then the function f is also differentiable.
![Then the function f is also continuous.
·-> |(x)ƒ − (x)uf [90] ⇒ A N <UA 0 < NE 0 <³A](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc30102f3-a996-44b5-a478-64ed6db32292%2F7c12d4fb-4d1a-4d0b-9d9d-e1bd89919caa%2Fcsjub_processed.png&w=3840&q=75)
Transcribed Image Text:Then the function f is also continuous.
·-> |(x)ƒ − (x)uf [90] ⇒ A N <UA 0 < NE 0 <³A
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