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Let's embark on the journey with one of the fundamental concepts in calculus: the continuous function. Imagine seamlessly drawing a curve without lifting your pencil; that's what it's like to trace a continuous function. Specifically, a continuous function is one where small changes in the input result in small changes in the output, creating a smooth graph. For instance, if you see the function f is continuous on an interval I, it means you can smoothly trace the function's graph from start to end over that interval without any breaks or jumps.

Continuity is vital for understanding Darboux sums and Riemann sums because it ensures we can find points within any subinterval of I where the function attains its minimum and maximum values—thanks to the Extreme Value Theorem. This characteristic is what bridges the gap between Darboux and Riemann sums for continuous functions: every infimum and supremum of subintervals can be matched with actual function values, making every Darboux sum a potential Riemann sum.

Next, let's delve into the increasing function. When we say a function is increasing on an interval I, imagine hiking up a hill that only goes upwards and never dips down. Mathematically, for any two points x and y in I where x < y, the value of the function at y (f(y)) is greater than at x (f(x)).

For such functions, when we partition the interval I and look at each piece, the left end will always have a lower function value than the right end. Hence, when constructing Darboux sums, the lower sum uses the function values at left endpoints, while the upper sum uses the values at right endpoints. In doing so, we automatically create Riemann sums since these endpoint values are valid sample points by definition. For students, this implies a direct relationship between increasing functions and their sums, making it easier to compute both upper and lower Darboux sums with certainty that they represent Riemann sums.

The Extreme Value Theorem is like the assurance that every contest has a winner and a loser. It states that if a function f is continuous over a closed interval, then f must reach a highest (maximum) and lowest (minimum) value at least once within that interval.

Why does this matter for our Darboux and Riemann sums discussion? Because it guarantees that within any subinterval of our partitioned interval I, no matter how small, our continuous function f will achieve its highest and lowest values somewhere within that subinterval. These 'extreme' values are what we use to construct our Darboux sums, knowing that they correspond to actual values that the function takes on, making it possible for these sums to align with the definition of Riemann sums, where any sample point is allowed.

Imagine slicing a baguette into pieces of various lengths—that's similar to partitioning an interval on the number line. A partition divides the interval I into smaller, non-overlapping subintervals, and each 'slice' is part of the whole. When it comes to Darboux sums and Riemann sums, the partition plays a vital role. The finer the partition (more slices), the closer these sums get to the actual area under the curve of the graph of our function f.

For a continuous or increasing function on interval I, we sample points within these sliced sections to compute the sums. With continuity on our side, those selected sample points will ensure that our Darboux sums align with Riemann sums, accurately estimating the area under f.

Let's round off our conceptual feast with the 'bookends' of sets: infimum and supremum. Picture a set of numbers; the infimum is the greatest number that is less than or equal to all numbers in the set, while the supremum is the smallest number that is greater than or equal to all numbers in the set. Think of them as anchors that bound the set from below and above.

In the context of Darboux sums, for each subinterval of our partition, we seek the infimum and supremum of our function's values. These are crucial because they form the lower and upper Darboux sums, respectively. What's the connection to Riemann sums, you ask? If our function is continuous or increasing on interval I, these infimum and supremum values can be pinpointed to actual function values, which allows us to also view them as possible Riemann sums. It's like matching the lowest and highest points of a terrain to actual coordinates on the map.