Question

Find a step function \(s(x)\) that approximates \(f(x)=x^{2}\) uniformly to within \(\varepsilon>0\) on \([0,1]\), in the sense that \(|f(x)-s(x)|<\varepsilon\) cierywhere in \([0,1]\).

Short answer

A step function which approximates \(f(x)=x^2\) uniformly to within \(\varepsilon>0\) on \([0,1]\) can be constructed by partitioning the interval into sufficiently small subintervals of width \(\Delta x = \sqrt\varepsilon\) and then defining the step function to be \(s(x) = \bar{x}^2\) on each subinterval, where \(\bar{x}\) is any number in that subinterval. The deviation between \(f(x)\) and \(s(x)\) will be less than \(\varepsilon\).

Step-by-step solution

Define the function

Start by defining the given function \(f(x)=x^2\) and the boundary interval \([0,1]\).

Partition the interval

Next, partition the interval \([0,1]\) into subintervals of width \(\Delta x\). The value of \(\Delta x\) is determined by the maximum deviation allowed, \(\varepsilon\). It is crucial to ensure that \(\Delta x\) is small enough so that the maximum difference between \(f(x)\) and any constant \(\bar{x}^2\) within each subinterval \([x-\Delta x, x]\) doesn't exceed \(\varepsilon\). It will depend on the specific number \(\varepsilon\) but a reasonable choice could be \(\Delta x = \sqrt{\varepsilon}\).

Define the step function

Now, define the step function \(s(x)\). Set \(s(x) = \bar{x}^2\) on each subinterval, where \(\bar{x}\) is any number in the subinterval. It can be chosen as the midpoint of each subinterval or the left or right endpoint - as long as it is inside the subinterval. Within each subinterval, the step function resolves to a single constant value. Since \(\Delta x\) is sufficiently small, the value of \(s(x)\) should not deviate from \(f(x)\) by more than \(\varepsilon\) in each subinterval.

Uniform approximation

Ensure that \(s(x)\) uniformly approximates \(f(x)=x^2\), in the sense that \(|f(x)-s(x)|<\varepsilon\) for every \(x\) in \([0,1]\). By construction, the maximum difference occurs at the boundaries of the subintervals and doesn't exceed \(\varepsilon\), thus \(s(x)\) is a successful approximation.

Key concepts

Step Function

A step function is a type of piecewise function that remains constant within each of its intervals. It "steps" up or down from one interval to the next without any gradual transition, resembling a staircase. The importance of step functions lies in their ability to provide simple approximations to more complex functions over a specific interval.
We use a step function to piece together constant values that best represent a curve within designated partitions. Step functions can be especially useful in numerical methods for approximating values when more precise calculations are not necessary or are computationally expensive.

• **Constant value:** For each segment of a step function, you assign a fixed value — typically the value of the function at a specific point within that segment.
• **Application in this exercise:** Here, the step function \( s(x) \) approximates \( f(x) = x^2 \) by choosing a constant value \( \bar{x}^2 \) within each subinterval, with \( \bar{x} \) commonly chosen as the midpoint. This approach minimizes discrepancies over the small segments.

Partition of Interval

Effective approximation of a function using a step function relies heavily on how we partition the interval. The interval in question — for this exercise, \[0,1\] — is broken into smaller segments so that within each part, the variation of the function \( f(x) = x^2 \) can be closely matched by a constant value.
The selection of \( \Delta x \), the width of each partition, plays a crucial role and is dictated by the level of accuracy desired, \( \varepsilon \). A finer (smaller \( \Delta x \)) partition results in a closer approximation.

• The smaller the width \( \Delta x \), the closer the step function can mimic the curve of \( f(x) \).
• One potential choice is \( \Delta x = \sqrt{\varepsilon} \), balancing the desired accuracy with computational efficiency.
• Each subinterval should ensure that the difference between \( f(x) \) and \( s(x) \) is kept under \( \varepsilon \).

Maximum Deviation

When approximating a function with a step function, maximum deviation quantifies how far the approximation can stray from the original function at any point within the interval. This deviation, \( |f(x) - s(x)| \), must be less than the specified \( \varepsilon \) to ensure a good approximation.
A function \( f(x) \) like \( x^2 \), which changes smoothly over \([0,1]\), can be best matched by ensuring that each partition is narrow enough to keep these deviations within bounds.

• The deviation is checked at boundaries of each subinterval since these are points where stepwise changes occur.
• Keeping \( \Delta x \) small enough inherently controls maximum deviation, fostering a closer fit of the approximation.
• The goal is to consistently keep deviation under control across all partitions, making sure that \( s(x) \) accurately reflects \( f(x) \) within each segment.

Boundary Interval

The focus on boundary intervals in this context is crucial, as they are where transition between steps in the step function occurs. It is essential to ensure that the approximation holds true across these points, which might otherwise witness major discrepancies.
Boundary intervals highlight the importance of evaluating the fitting accuracy for \( s(x) \) at transition points, assuring that each step accurately follows the curve of \( f(x) = x^2 \) within the set tolerance.

• Boundaries are the edges of each subinterval, points like 0, \( \Delta x \), 2\( \Delta x \)... up to 1 in the interval \([0,1]\).
• We verify that the step function remains within \( \varepsilon \) of the original function consistently across these boundaries.
• A careful choice of partition \( \Delta x \) helps in maintaining continuity in the approximation, preventing any large jumps at these critical points.