Proving Continuity

**okor** · October 31st 2010, 05:09 PM

I have the function $f<img src=$ (a,b),U_{(a,b)})\rightarrow((c,d),U_{(c,d)})" alt="f

(a,b),U_{(a,b)})\rightarrow((c,d),U_{(c,d)})" />, where $U_{(a,b)}$ is the subspace topology of the usual topology where A = (a,b) and $U_{(c,d)}$ is the subspace topology of the usual topology where A = (c,d), defined as $f(x)=\frac{d-c}{b-a}x+c-\frac{a(d-c)}{b-a}$

How do I show this function is continuous?

**Tinyboss** · October 31st 2010, 06:24 PM

It's a polynomial.

**okor** · October 31st 2010, 06:28 PM

Originally Posted by Tinyboss

It's a polynomial.

I know all polynomials are continuous everywhere, but I have to actually show it is continuous, I can't just say it is continuous because it is a polynomial.

**Tinyboss** · October 31st 2010, 06:37 PM

Why are polynomials continuous? It comes down to sum and product behavior of limits. Of course, this polynomial is actually linear, so you can probably hammer out an epsilon-delta just for this one case if you prefer.

**okor** · October 31st 2010, 06:50 PM

Originally Posted by Tinyboss

Why are polynomials continuous? It comes down to sum and product behavior of limits. Of course, this polynomial is actually linear, so you can probably hammer out an epsilon-delta just for this case if you prefer.

So for some $(y_{1},y_{2})$ in $U_{(c,d)}$ , as $y_{1}\rightarrow c^{+}$ and $y_{2}\rightarrow d^{-}$ , $f^{-1}((y_{1},y_{2}))\rightarrow (a,b)$ , thus $f^{-1}((y_{1},y_{2}))$ is in $U_{(a,b)}$ .

Would something like that work/be accurate?

Thread: Proving Continuity

LinkBack

Thread Tools

Display