# Proving Continuity

• Oct 31st 2010, 05:09 PM
okor
Proving Continuity
I have the function $\displaystyle f:((a,b),U_{(a,b)})\rightarrow((c,d),U_{(c,d)})$, where $\displaystyle U_{(a,b)}$ is the subspace topology of the usual topology where A = (a,b) and $\displaystyle U_{(c,d)}$ is the subspace topology of the usual topology where A = (c,d), defined as $\displaystyle f(x)=\frac{d-c}{b-a}x+c-\frac{a(d-c)}{b-a}$

How do I show this function is continuous?
• Oct 31st 2010, 06:24 PM
Tinyboss
It's a polynomial.
• Oct 31st 2010, 06:28 PM
okor
Quote:

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.
• Oct 31st 2010, 06:37 PM
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 one case if you prefer.
• Oct 31st 2010, 06:50 PM
okor
Quote:

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

Would something like that work/be accurate?