Proof of Continuity: Solving Problem a and b

[asif zaidi]

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 $$\alpha,$$$$\alpha$$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

 

[sutupidmath]

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!

 

[HallsofIvy]

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 $$\alpha,$$$$\alpha$$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 "$\epsilon, \delta$ 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, $\lim{x\rightarrow z} f(x)= f(z)$ and since g is continuous at z, ...", using that to show that $\lim{x\rightarrow z} f(x)+ g(x)= f(z)+ g(z)$ 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