- #1
asif zaidi
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Problem statement
Given f:[1,2]->R defined by f(x) = x^2. Show that this function is continuous
Problem Solution (my version at least)
1- It is known sequence {a[n]b[n]} converges to {ab}
2- Definition of continuous: if every sequence {cn} in f we have f(cn) -> f(c)
3- Given our domain of 1<= f <=2 assume our sequence an-> a=1 and bn-> b=2
4- Therefore f(anbn) = f(1*2) = f(a*b) = f(1*2) = 4
5- Therefore this function is continuous.
My concern is am I taking a leap of faith in 3,4 which is the essence of my proof.
Is there a better way to prove this or is this mathematically sufficient.
Thanks
Asif
Given f:[1,2]->R defined by f(x) = x^2. Show that this function is continuous
Problem Solution (my version at least)
1- It is known sequence {a[n]b[n]} converges to {ab}
2- Definition of continuous: if every sequence {cn} in f we have f(cn) -> f(c)
3- Given our domain of 1<= f <=2 assume our sequence an-> a=1 and bn-> b=2
4- Therefore f(anbn) = f(1*2) = f(a*b) = f(1*2) = 4
5- Therefore this function is continuous.
My concern is am I taking a leap of faith in 3,4 which is the essence of my proof.
Is there a better way to prove this or is this mathematically sufficient.
Thanks
Asif