I am having trouble with the following proofs. If someone can help I would appreciate it.(adsbygoogle = window.adsbygoogle || []).push({});

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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# Proof of Continuity

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