A function f on [0,1] is continuous everywhere but has a sharp corner — it is not differentiable at one point. According to the Weierstrass Approximation Theorem, which of the following is true?
Af cannot be uniformly approximated by polynomials because polynomials are smooth and f has a corner
Bf can be uniformly approximated by polynomials because only continuity is required
Cf can only be approximated pointwise, not uniformly, because of the non-differentiable point
DThe theorem does not apply since f fails to be smooth
The Weierstrass Approximation Theorem requires only continuity on a closed interval — not differentiability, smoothness, or any other regularity condition. The most tempting wrong answer (option A) reflects the misconception that since polynomials are infinitely differentiable, they can only approximate functions of similar smoothness. But uniform approximation does not require matching derivatives — it only requires making |f(x) − P(x)| small for all x simultaneously. A corner poses no barrier.
Question 2 Multiple Choice
What does it mean to say polynomials are 'dense' in C([a,b]) — the space of continuous functions on [a,b] with the uniform norm?
AEvery continuous function is itself a polynomial
BEvery continuous function is the limit of polynomials in the pointwise sense at every x
CEvery continuous function can be approximated arbitrarily closely by some polynomial in the sup-norm
DPolynomials make up more than half the functions in C([a,b])
Density in the uniform (sup-norm) topology means that for any f ∈ C([a,b]) and any ε > 0, there exists a polynomial P with sup_{x∈[a,b]} |f(x) − P(x)| < ε. This is precisely what 'uniformly approximated' means. Option B describes pointwise convergence, which is weaker — it allows the approximation error to depend on x. The theorem guarantees the stronger, uniform version: the worst-case error over the entire interval can be made as small as desired.
Question 3 True / False
The Weierstrass Approximation Theorem guarantees that for any ε > 0, there exists a polynomial P such that |f(x) − P(x)| < ε holds simultaneously for all x in [a,b], not just at individual points.
TTrue
FFalse
Answer: True
This is exactly the content of the theorem — the approximation is uniform, meaning the same error bound holds across the entire interval at once. This is stronger than pointwise approximation, where ε could depend on x. The Bernstein polynomial construction achieves this: for each n, Bₙ(f, x) approximates f at every x, and as n → ∞ the sup-norm distance to f goes to zero.
Question 4 True / False
Because polynomials are infinitely differentiable, the Weierstrass Approximation Theorem applies mainly to continuous functions that are also differentiable on (a,b).
TTrue
FFalse
Answer: False
The theorem's hypothesis is continuity on the closed interval [a,b] — differentiability is not required. The appeal of this misconception is that polynomials are smooth, so it seems they should only approximate smooth things. But uniform approximation is about how close function values are, not about matching derivatives. A continuous but nowhere-differentiable function (like the Weierstrass function itself!) can still be uniformly approximated by polynomials.
Question 5 Short Answer
Why is it surprising that polynomials can uniformly approximate every continuous function on [a,b], and why does the theorem require a closed (bounded) interval rather than all of ℝ?
Think about your answer, then reveal below.
Model answer: Polynomials are algebraically rigid — they grow without bound outside any bounded region and are determined by finitely many coefficients. An arbitrary continuous function can have complex local behavior. The surprise is that despite this rigidity, polynomials can track any continuous function to arbitrary precision on a compact interval. The closed interval is essential because polynomials diverge as x → ±∞, so uniform approximation over all of ℝ fails for any bounded function. Compactness allows the law-of-large-numbers argument in the Bernstein proof to control errors uniformly.
The compactness of [a,b] is doing real work: it ensures that the supremum of the error is actually attained (rather than approached), and that the Bernstein polynomials — which average f's values according to a binomial distribution — concentrate near any given x as n grows. On an unbounded domain, polynomial approximation can fail even for simple bounded continuous functions like sin(x).