MATH114 Integration and Differentiation

Problem Solving Class 1

[2.5ex]

Let $f:[a,b]\to ℝ$ be a continuous function. Let us write $\mathrm{\ell }(f,[a,b],n,\cdot )$ and $u(f,[a,b],n,\cdot )$ for the lower and upper approximating step functions, respectively, see Definition 1.2.11.

• What do you think why “approximating step functions” are called the way they are?

• Draw the graph of

  $$f:(-1,1)\to ℝ,f(x)=x^2.$$

  Include the graphs of the approximating step functions $u(f,[a,b],n,\cdot )$ and $\mathrm{\ell }(f,[a,b],n,\cdot )$ for $n=1,2,3$. What do they approximate? Can you tell from the graph in what sense they approximate it?

• Consider the function

  $$g:(-1,1)\to ℝ,g(x)=0,$$

  and also the following two sequences of functions ${(g_n)}_{n\in \mathbb{N}}$ and ${({\tilde{g}}_n)}_{n\in \mathbb{N}}$ (rather than the usual sequences of real numbers) defined as follows:

  $$g_n:(-1,1)\to ℝ,g_n(x)=\frac{1}{n}{x}^2$$

  and

  $${\tilde{g}}_n:(-1,1)\to ℝ,{\tilde{g}}_n(x)=x^n.$$

  Explain if and in what sense the two sequences ${(g_n)}_{n\in \mathbb{N}}$ and ${({\tilde{g}}_n)}_{n\in \mathbb{N}}$ converge to $g$. Which one would you call “pointwise convergent” and which one “uniformly convergent”? Why?

• Using the intuition from step 3, can you suggest a general formal definition of what it means for a sequence of functions to be “uniformly convergent” to some other function (without looking up the proper definition on Wikipedia)? Compare your suggestions.

• Can you now formulate a precise statement of how the approximating step functions $\mathrm{\ell }(f,[a,b],n,\cdot )$ and $u(f,[a,b],n,\cdot )$, $n\in \mathbb{N}$, approximate $f$? Make a conjecture based on steps 2 and 4.

• Prove your conjecture in step 5.