Show that a continuous function $f$ can in fact continuously deformed (homotopy) into an arbitrary close $C^{\infty}$-function : There exist a continuous function ("homotopy")$ H:I\times I \rightarrow \mathbb{R}$ s.t $H(t,0)=f(t) ,t\in I $ and s.t $H(.,s)$ is a class of $C^{\infty}$ for each fixed s, $0<s\leq1$ by uniform continuity (!) of $H$ , $H(.,s)$ will be arbitrarily close to $f=H(.,0)$ provided s is sufficiently small. $\textbf{I have a hint for this to solve as given}$
Prove this by analyzing the known proof with the polynomial kernel $(1-x^2)^{n}$. The integer exponents n are used to obtain a $\textbf{discrete}$ approximation ( by polynomials ) . How to get a continuous approximation instead ( not necessarily with polynomials )
Please someone Can elaborate that problem in details . I have a hint that I can use $(1-x^2)^{1/s}$ instead of using the kernel $(1-x^2)^{n}$ But I am unable to get the clear idea how it will work . I will either get a polynomial approximation or something else .
If someone can suggest me some reference book to understand it I will be very thankful for it .