In my math textbook they have a proof that an antiderivative to f(x) is the same as the area function to f(x). They assume that f(x) is continuous and that f(x) is a growing function. By looking at the pictures above and drawing some rectangles they come to the following result:
Where $A(x_0+h)-A(x_0)$ is the area of the yellow part under the curve of figure 702c. $ f(x_0)*h$ is the area of the small rectangle, and $f(x_0+h)*h$ is the area of the large rectangle.
But since f(x) is a growing function we know for certain that: $f(x_0)*h < f(x_0+h)*h $
So my question is, why do they write $ f(x_0)*h ≤ A(x_0+h) - A(x_0) ≤ f(x_0+h)*h$
When in reality they should write $f(x_0)*h < A(x_0+h)-A(x_0) < f(x_0+h)*h $
and if that's the case, does the proof still work?
Edit
The proof then goes on like this:
$ f(x_0) < \frac{A(x_0+h)-A(x_0)}{h} < f(x_0+h) $
If we let h approach zero we end up with: 
We then finally get:
$ f(x_0) < A'(x_0) < f(x_0) $
And isn't that result that an anomaly?
Yes, the proof still works if $f(x)$ is a growing function.
However, if we want to generalize to ANY function, then I think the $<=$ is necessary. The point is that the value of the area always lies BETWEEN those two bounds, which can be easily generalized if we use the $<=$ sign.
For example, for a generalized function consider $f(x)=2$ In this case, all three bounds will be equal, i.e. $f(x_0)*h = A(x_0 + h) - A(x_0) = A(x_0+h)$