Question

How can we prove mathematically that a root function is a continuous function?

Answer 1

Does it suffice to prove that it's uniformly continuous? My continuity proofs are a bit rusty :-(

Answer 2

Anyway, I think you can prove it with monotonic increase and the fact that the range is an interval.

Answer 3

please make me understand by a example.

Answer 4

Well, there's a theorem that states that if a function f is monotic and its range is an interval, then the function is continuous; so, it's a straightforward application of the theorem.

Answer 5

monotonic*

Answer 6

Theorem 3.2.5 http://www.math.utah.edu/~taylor/3_Continuous If you need the proof for it.

Answer 7

how to prove f(x) =\[\sqrt{2x} +1 is continuous\]

Answer 8

Well, it's monotonic and its range is an interval, so you can apply the above mentioned theorem. Proving that a given function is continuous, in general, is hard, as we are dealing with epsilon-delta stuff. But, like I said, I am not up to it, it has been more than a year since I last did any exercises on this stuff.

Answer 9

np .:) & only last question, what is monotonic?please tell.

Answer 10

For every x <= y, then f(x) <= f(y) represents a monotonic function. I can give you a proof for \(\sqrt{x} \) continuity, but that's all.

Answer 11

monotonically increasing, that is. Decreasing is similar, only inverted.

Answer 12

k! thanx a lot for co-operating me.

Answer 13

A proof based on the Mean Value Theorem would go something like this: \[ \lim_{x \rightarrow 0} \sqrt{x} = 0 \]and \[ \lim_{x \rightarrow a} \sqrt{x} = \sqrt{a}\]So, sqrt(x) is continuous. The second limit, tho, we have to prove it using the epsilon-delta definition.

Answer 14

And there is a typo there, should be \( \lim_{x \rightarrow 0^+} \) for the first limit.

Answer 15

k! thanx i got that what u told

Answer 16

No problem, and sorry for not being able to help you more :-)

Answer 17

its ok! but u really helped me in the best way :)