1. .. ..

2. Come to think of it maybe some kind of parabola or something where it changes direction and areas cancel out?

3. Did you type the question correctly or am I missing something ? There are several functions which are continuous over [-1,2] with partitions of that same interval used to calculate the areas under them.

4. Well right, the general question was do they exist? if so give an example if not why not, at first I didn't think they existed but now im just thinking of examples that fit that.

5. Any linear equation would work really. For example, consider $y = -x + 3$ over the interval [-1, 2].

Length of each subinterval:
$\Delta x = \frac{2 - (-1)}{n} = \frac{3}{n}$

Consider your partition with left sample points:
$x_{0} = - 1 \:\: < \:\: x_{1} = -1 + \frac{3}{n} \:\: < \:\: x_{2} = -1 + 2\cdot \frac{3}{n} \:\: < \:\: ... \:\: <$ ${\color{blue}x_{i} = -1 + \frac{3i}{n}} \:\: < \:\: ... \:\: < \:\: x_{n} = -1 + \frac{3n}{n} = 2$

$\lim_{n \to \infty}\sum_{i = 1}^{n} f\left(x_{i}^{*}\right) \Delta x$
$x_{i}^{*} = x_{i}$