1. ## Proving Continuity

I have the function $f(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?

2. It's a polynomial.

3. 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.

4. 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.

5. 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?