Lipschitz Conditions for Inverses: Sqrt Example

Suppose we start with positive real numbers $0<A<B$. Let $f(x)=x^{2}$ on the interval $[A,B]$:


sqrtlip1.wmf
We know that $f$ satisfies the following Lipschitz conditions on the interval $[A,B]$:MATHso, by the Inverse Function Theorem, its inverse, the square-root, satisfies:MATHon the interval $[A^{2},B^{2}]$. If we let $C=A^{2}$ and $D=B^{2}$, then this Lipschitz condition becomes:MATH

Suppose, now, that we had started with the interval $[C,D]$, where $0<C<D$ and numbers $z\leq w$ in this interval. We'd like to say the same Lipschitz conditions hold, but how do we know that there are numbers $A$ and $B$ whose squares are $C$ and $D$ respectively. In other words, how do we know that $C$ and $D$ have square roots? Clearly it suffices to show that $C>0$ has a square root, since the same argument will show that $D$ has one. The idea is to show that there are numbers $0<U<V$ such that $U^{2}<C<V^{2}$ since then we can apply the Inverse function theorem to $(\ \ )^{2}$ on the interval $[U,V]$ (with $z=C$). We will use the inequality MATHeasily proved by induction. When $N=2$ this becomes $(1+v)^{2}\geq 1+2v$. We will let $V=1+v$ and choose $v$ so that $1+2v>C$. This will happen when $v>\dfrac{C-1}{2}$, so we can let $v=C+1$, which givesMATHLetting $V=1+v$ ($=C+2$) now gives $V^{2}>C$. To construct $U$ such that $U^{2}<C$, use the procedure just described to find a number $Y$ such that $Y^{2}>\dfrac{1}{C}$. Then MATH; thus, we let $U=\dfrac{1}{Y}$.




Exercise 1. Find numbers $U$ and $V$ such that MATH. Explain in detail why we can now conclude that $\dfrac{2}{7}$ has a square root.

Exercise 2. If $0<C<D$, prove that the square root function satisfies the following Lipschitz condition on the interval $[C,D]$:MATH



Exercise 3. Explain how to generalize this discussion to find Lipschitz conditions for the $N$th root function $z^{1/N}$ on a positive interval $[C,D]$.



Exercise 4. Suppose $A>1$. For any $C>0$, use MATH to find an exponent $V$ such that $A^{V}>C$. Also, find an exponent $U$ such that $A^{U}<C$. (Hint: Let $\alpha =A-1>0$, so that MATH; imitate what we did for squaring.)



Exercise 5. Deduce from Exercise 4 that any number $z>0$ has a logarithm to base $A$ ($A>1$). You'll need the inverse function theorem applied to $A^{x}$.



Exercise 6. Find Lipschitz conditions for $\log _{A}($ $)$ on the positive interval $[C,D]$.