Approximating not-nice functions by nice functions

In the last class I asserted that it was possible to approximate certain not-nice functions like and $\mid t\mid$ by $B$-nice functions. For example, I asserted that for all $t_0\in ℝ$ and there exists a function ${\Delta }_{t_0,\lambda }:ℝ\to ℝ$ which is $O(1/{\lambda }^4)$-nice and approximates the $t_0$-step-function in the following sense: ${\Delta }_{t_0,\lambda }\left(t\right)=1$ for ; ${\Delta }_{t_0,\lambda }\left(t\right)=0$ for $t>t_0+\lambda$; and, $0\le {\Delta }_{t_0,\lambda }\left(t\right)\le 1$ for $\mid t-t_0\mid \le \lambda$.

There was a question in class as to whether this could really be possible. Specifically, if ${\Delta }_{t_0,\lambda }$ is $0$ for all $t>t_0+\lambda$, and it's smooth, then it has all derivatives equal to $0$ for all $t>t_0+\lambda$. Shouldn't this make it $0$ everywhere?

Well, not so. It's true that for any $t>t_0+\lambda$, all derivatives are $0$. But Taylor's theorem does not force such a function to be $0$ everywhere. Remember that Taylor (for smooth functions) just says that for any $r\in \mathbb{N}$, $f(x+\epsilon )=f\left(x\right)+\epsilon fʹ\left(x\right)+\frac{{\epsilon }^2}{2!}fʺ\left(x\right)+\cdots +\frac{{\epsilon }^{r-1}}{(r-1)!}{f}^{(r-1)}\left(x\right){\epsilon }^{r-1}+\frac{{\epsilon }^r}{r!}{f}^{\left(r\right)}\left(y\right)$ for some $y\in [x,x+\epsilon ]$.

For a concrete example, consider the function $\phi \left(t\right)$ which is $\exp (-1/(1-t^2))$ for , and is $0$ for $\mid t\mid \ge 1$. It's easy to see that $\phi \left(t\right)$ is smooth on $(-1,1)$ and it's not hard to check that its derivatives, of all orders, at $\pm 1$ (when approached from inside) are $0$.

In fact, functions like $\phi$ (known as bump functions or mollifiers) are what one uses to create functions like ${\Delta }_{t_0,\lambda }$ -- essentially, by taking $\Delta *{\phi }_{\lambda }$, where ${\Delta }_{t_0}$ denotes the actual discontinuous step function, $*$ denotes convolution, and ${\phi }_{\lambda }$ denotes a compressed version of $\phi$, viz., ${\phi }_{(}t/\lambda )/\lambda$.