Approximating continuous function with polynomials

Question

Given $f \colon[a,b] \to \mathbb{R}$ such that $f$ is $N$ times continuously differentiable (so $f$ itself is continuous) and by the Weierstrass approximation theorem, for $\epsilon >0$ there is a polynomial $p$ such that for all $x \in [a,b]$ we get that $|f(x)-p(x)| < \epsilon$. The exercise asks to find a polynomial $p$ such that for all $1 \le k \le N$ and for all $x \in [a,b]$ we get that $|f^{(k)} (x) - p^{(k)} (x) | < \epsilon$, i think the same polynomial works but I don't know how to prove it even for $|f'(x) -p'(x)| < \epsilon$.

Answer 1

Even if a polynomial $p$ approximates a function $f$ well, $p'$ might not approximate $f'$ well: For example, if $f=0$ on $[0,1]$ and $p_n(x)=\frac{x^n-x}{\sqrt{n}}$, then $$\Vert f-p\Vert_\infty\leq 2/\sqrt{n},$$ but $$\Vert f'-p'\Vert_\infty\geq n-1$$

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As for the solution of your exercise, use Weierstrass to approximate $f^{(N)}$ by a polynomial $p$, and then prove, using the fundamental theorem of calculus, that the polynomial $f^{(N-1)}(a)+\int_a^x p(t)dt$ approximates $f^{(N-1)}$ well enough and iterate. You should check the details yourself.