1. **State the problem:** We need to find a one-to-one correspondence (bijection) function from set $A=[0,1]$ to set $B=[1,2]$. 2. **Recall the definition:** A function $f:A \to B$ is a one-to-one correspondence if it is both injective (one-to-one) and surjective (onto). 3. **Construct a candidate function:** Consider the function $$f(x) = x + 1$$ for $x \in [0,1]$. 4. **Check injectivity:** If $f(x_1) = f(x_2)$, then $x_1 + 1 = x_2 + 1$ implies $x_1 = x_2$. So $f$ is injective. 5. **Check surjectivity:** For any $y \in [1,2]$, we can find $x = y - 1 \in [0,1]$ such that $f(x) = y$. So $f$ is surjective. 6. **Conclusion:** The function $$f(x) = x + 1$$ is a one-to-one correspondence from $[0,1]$ to $[1,2]$. **Final answer:** $$f(x) = x + 1$$