Use differentiable function to approximate continuous function

Question

Let $f$ be a continuous monotone increasing function on $[a,b]$. I want to show for any $\epsilon>0$ there exist a monotone increasing function $g\in C^1[a,b]$ such that $$ \max_{x\in[a,b]}|g-f|<\epsilon $$ Could anyone give me some hint? Thanks in advance

Answer 1

By looking at $g(x)=\frac {f(x)-f(a)} {f(b)-f(a)}$ for $a<x<b$, $1$ for $x \geq b$ and $0$ for $x \leq a$ we can reduce this problem to the case when $f$ is a CDF on $\mathbb R$. Can we approximate a continuous CDF by as continuously differentiable CDF? Probabilistically this is very simple. Let $X$ be a r.v. with CDF $F$. Let $Y $ have some nice CDF with a continuous density, say standard normal CDF. Let $F_n$ be the distribution function of $X+\frac Y n$. Then $F_n$ has a continuous density $F_n'$ and $F_n \to F$ uniformly on $\mathbb R$.

[One way to show that $F_n$ is continuously differentiable is to use the inversion foumula for charactersitic functions. Note that the charactersitic functions of $X+\frac Y n$ is integrable. You can also prove this by just Calculus by writing $F_n$ as convolution of two CDF's]

Answer 2

W.l.o.g. let $[a,b]=[0,1]$ and let $f \in C([0,1])$. The Bernstein Polynomials $$ B_n(x)=\sum_{k=0}^n {n \choose k} f(k/n)x^k(1-x)^{n-k} $$ have the derivative $$ B_n'(x)=\sum_{k=0}^n {n \choose k}f(k/n)(kx^{k-1}(1-x)^{n-k}-(n-k)x^k(1-x)^{n-k-1}) $$ $$ =n\sum_{k=0}^{n-1} {n-1 \choose k} [f((k+1)/n)-f(k/n)] x^k(1-x)^{n-1-k}. $$ If $f$ is increasing then clearly $B_n'(x) \ge 0$ on $[0,1]$ and $(B_n)$ is uniformly convergent to $f$.