Proof of Continuity: Solving Problem a and b

In summary: YIOu can either use directly epsylon, delta definition of continuity of a function, or you can approach it this way:since f, g continuous at z, we have\lim_{x\rightarrow\ z}f(x)=f(z) \ \ ,\ \lim_{x\rightarrow\ z} g(x)=g(z) now\lim_{x\rightarrow\ z}(f(x)+g(x))= \lim_{x\rightarrow\ z}f(x)+\lim_{x\rightarrow\ z} g(x)=f(a)+g(a) which actually is what we want to show!
  • #1
asif zaidi
56
0
I am having trouble with the following proofs. If someone can help I would appreciate it.

Problem Statement

Given that f, g are continuous at z, prove that

a- f+g is continuous at z
b- For any complex [tex]\alpha, [/tex][tex]\alpha[/tex]f is continuous at z

There are other parts to this but if I think if I can get help on a) I think the rest will follow


Solution


The definition of continuous function is
(lim x->c) f(x) = f(c)

I have to determine if f+g is continuous at z. To do this I am proceeding as follows:

1. (lim x->z) (f+g) = (lim x->z) f(x) + (lim x->z) g(x)
2) Since (lim x->z) f(x) = continuous (given) and g(x) is coninuous (given) the sum will also be continuous.

Is this mathematically sufficient or am I missing a step.
My problem with proofs is that I use the result in my explanation which is a no-no.

Any help will be appreciated.

Thanks

Asif
 
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  • #2
YIOu can either use directly epsylon, delta definition of continuity of a function, or you can approach it this way:
since f, g continuous at z, we have
[tex]\lim_{x\rightarrow\ z}f(x)=f(z) \ \ ,\ \lim_{x\rightarrow\ z} g(x)=g(z) [/tex] now

[tex]\lim_{x\rightarrow\ z}(f(x)+g(x))= \lim_{x\rightarrow\ z}f(x)+\lim_{x\rightarrow\ z} g(x)=f(a)+g(a)[/tex] which actually is what we want to show!
 
  • #3
asif zaidi said:
I am having trouble with the following proofs. If someone can help I would appreciate it.

Problem Statement

Given that f, g are continuous at z, prove that

a- f+g is continuous at z
b- For any complex [tex]\alpha, [/tex][tex]\alpha[/tex]f is continuous at z

There are other parts to this but if I think if I can get help on a) I think the rest will follow


Solution


The definition of continuous function is
(lim x->c) f(x) = f(c)

I have to determine if f+g is continuous at z. To do this I am proceeding as follows:

1. (lim x->z) (f+g) = (lim x->z) f(x) + (lim x->z) g(x)
So you can use that and are not required to go back to "[itex]\epsilon, \delta[/itex] to prove it?

2) Since (lim x->z) f(x) = continuous (given) and g(x) is coninuous (given) the sum will also be continuous.
Isn't that simply a restatement of the theorem you want to prove? No, not exactly- I just reread it and while you just say "g(x) is continuous", you say "(limx->z)f(x)= continuous" but I can't make heads or tails of that. Surely the limit is not the word "continuous"! Perhaps you mean to say, "Since f is continuous at z, [itex]\lim{x\rightarrow z} f(x)= f(z)[/itex] and since g is continuous at z, ...", using that to show that [itex]\lim{x\rightarrow z} f(x)+ g(x)= f(z)+ g(z)[/itex] and then using that to show that f+ g is continuous. If your teacher is a real hard nose, you might need to show what "f(x)+ g(x)" has to do with "f+ g"!

Is this mathematically sufficient or am I missing a step.
My problem with proofs is that I use the result in my explanation which is a no-no.
It certainly is! I would strike you across the nose with a rolled up newspaper!

Any help will be appreciated.

Thanks

Asif
 

Related to Proof of Continuity: Solving Problem a and b

What is "Proof of Continuity"?

"Proof of Continuity" is a mathematical concept that refers to the property of a function being continuous. In simpler terms, it means that there are no sudden or abrupt changes in the behavior of the function, and it can be drawn without lifting the pen from the paper.

What is the importance of "Proof of Continuity" in solving problems?

"Proof of Continuity" is important because it allows us to analyze and understand the behavior of a function. It helps us to determine if a function is smooth and well-behaved, and to make accurate predictions and calculations based on the function's behavior.

How do you prove continuity of a function?

To prove continuity of a function, we need to show that the function is defined at a given point, the limit of the function at that point exists, and the limit is equal to the value of the function at that point. This can be done using various techniques, such as the epsilon-delta definition of continuity or the intermediate value theorem.

What are some common applications of "Proof of Continuity"?

"Proof of Continuity" has many practical applications, such as in physics, engineering, and economics. For example, it can be used to model and analyze the motion of objects, the behavior of electrical circuits, and the growth of populations. It is also crucial in optimization problems, where finding the maximum or minimum value of a function is necessary.

Can "Proof of Continuity" be used to solve real-world problems?

Yes, "Proof of Continuity" can be used to solve real-world problems. Many real-world phenomena can be described by continuous functions, and by proving the continuity of these functions, we can gain insights and make accurate predictions about the behavior of these phenomena. Additionally, many real-world problems can be formulated as optimization problems, which often require the use of continuity to find the optimal solution.

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