UIC Master’s Exam- Fall 2007 R2

Problem Statement: $f$ is continuous on $\mathbb{R}$. Given that $g(x)=\dfrac{1}{2\delta}\int\limits_{-\delta}^{\delta}f(x+t)dt$

(a) prove $g$ has continuous derivative on $\mathbb{R}$ (Hint: start with the change of variable u=x+t). (b) Given $[a,b]$ and $\varepsilon>0$ show that there exists a $\delta>0$ such that $\left|g(x)-f(x)\right|<\varepsilon$ for every $x\in [a,b]$.

Solutions:

(a)Proof: First let us start by making the change of variable suggested in the problem statement.

Let $u=x+t$, then $du=1dt$ $g(x)=\dfrac{1}{2\delta}\int\limits_{x-\delta}^{x+\delta}f(u)du$

Now we may apply the Fundamental Theorem of Calculus and obtain that $g'(x)=\dfrac{1}{2\delta}\left(f(x+\delta)-f(x-\delta)\right)$

which is continuous since $f$ is continuous on $\mathbb{R}$ $\Box$

(b)Proof: We are given $[a,b]$ and $\varepsilon>0$. Since $f$ is continuous on a compact set it is uniformly continuous, thus, for our given $\varepsilon$ we may find a $\delta>0$ such that $\left|f(x)-f(y)\right|<\varepsilon$ whenever $\left|x-y\right|<\delta$. Now consider the following: $\left|g(x)-f(x)\right|=\left|\dfrac{1}{2\delta}\int\limits_{-\delta}^{\delta}f(x+t)dt-f(x)\right|$ $=\left|\dfrac{1}{2\delta}\int\limits_{-\delta}^{\delta}\left(f(x+t)-f(x)\right)dt\right|$ $\leq \dfrac{1}{2\delta}\int\limits_{-\delta}^{\delta}\left|f(x+t)-f(x)\right|dt$

And so, for $\left|t\right|<\delta$ it follows that $\left|f(x+t)-f(x)\right|<\varepsilon$. This allows us to simplify further: $<\dfrac{1}{2\delta}\int\limits_{-\delta}^{\delta}\varepsilon dt$ $=\dfrac{1}{2\delta}\left(\varepsilon\delta-(-\varepsilon\delta)\right)$ $=\varepsilon$ $\Box$

Reflection: Always try the easiest method first!!! This is a lesson I keep “learning” but I don’t really seem to be learning it. Part (b) I attempted to do in a very strange way. Hopefully now I have really learned, try the easier method first!! If it doesn’t work then at least you’ll have gained some insight into why it isn’t working, and that can help you with another attempt.