Question

f(x) = x^(2/3)

Prove that it is continuous everywhere on [0,1]

Answer 1

Show that, either:
\[
\lim_{x\to c^+}x^{\frac{2}{3}}=\lim_{x\to c^-}x^{\frac{2}{3}}=c^{\frac{2}{3}}
\]For some \(c\in [0,1]\)

Or:
\[
\lim_{h\to 0^+}(x+h)^{\frac{2}{3}}=\lim_{h\to 0^+}(x-h)^{\frac{2}{3}}=x^\frac{2}{3}
\]\(\forall x \in \mathbb{R}\)

Answer 2

Can I say that
Lim x--> c+ f(x) = c^(2/3)
Lim x--> c- f(x) = c^(2/3)
So Lim x--> c f(x) = c^(2/3) which exists -----> but can i say it is finite?

Answer 3

Yes, and, yes, but I don't know what you mean by 'it is finite.'

Answer 4

As in c^(2/3) is not + or - infinity

Answer 5

Well, you can prove that, if \(a\ge 0\)
\[
a^3\ge a^2
\]Which implies:
\[
a\ge a^\frac{2}{3}
\]Clearly, \(a\) is non-infinite for finite values, hence, since \(a^\frac{2}{3}\) is smaller than \(a\) for all positive real values, it is non-infinite.

Answer 6

Using the lim h--> 0 f(x) way, can you show me how to do Lim h--> 0+ f(x) = Lim h--> 0- f(x)?

Answer 7

Actually, I think for the 2nd way, shouldn't it be Lim h--> 0+( f(x+h) - f(x))/h?

Answer 8

Nevermind! I figured it out

Answer 9

Sure:
\[
(x+h)^\frac{2}{3}=\sqrt[3]{x^2+2xh+h^2}
\]If we take the limit as \(h\to 0^\pm\):
\[
\sqrt[3]{x^2+2xh+h^2}=\sqrt[3]{x^2+2x\cdot0+0^2}=\sqrt[3]{x^2}=x^\frac{2}{3}
\]

Answer 10

No, that last part is a derivative, not a definition of continuity.

Answer 11

Wait or was that for derivatives?

Answer 12

In response to: "Actually, I think for the 2nd way, shouldn't it be Lim h--> 0+( f(x+h) - f(x))/h?"

Answer 13

If I'm doing limits, you were right, it should have been lim of f(x) not the slope?

Answer 14

That is correct.

Answer 15

What's the diff between the 2 ways?

Answer 16

They're exactly equivalent. Sometimes, it happens that one is easier to use than the other, but they're no different, mathematically.

Answer 17

Well, I'm off, hopefully that helped.

Answer 18

Thank you!

Answer 19

Quick question: the first way is general and the 2nd way is for a specific point?

Answer 20

I mean the 1st way is specific point and 2nd way is in general?

Answer 21

Yes.

Answer 22

Ok! Thanks!