# continuous function

• Oct 28th 2008, 10:44 AM
rmpatel5
continuous function
Let h: http://qaboard.cramster.com/Answer-B...4687502181.gif be continuous on http://qaboard.cramster.com/Answer-B...6562508596.gif satisfying http://qaboard.cramster.com/Answer-B...0937501839.gif=0 for all m within integers and n within naturals. Show that h(x)=0 for all x within reals
• Oct 28th 2008, 11:44 AM
ThePerfectHacker
Quote:

Originally Posted by rmpatel5
Let h: http://qaboard.cramster.com/Answer-B...4687502181.gif be continuous on http://qaboard.cramster.com/Answer-B...6562508596.gif satisfying http://qaboard.cramster.com/Answer-B...0937501839.gif=0 for all m within integers and n within naturals. Show that h(x)=0 for all x within reals

Let $\displaystyle r\in \mathbb{R}$ with $\displaystyle h(r) > 0$ WLOG. Take $\displaystyle \epsilon > 0$ so that $\displaystyle h(r) - \epsilon > 0$. Then it means there is $\displaystyle \delta > 0$ such that if $\displaystyle x\in (r-\delta,r+\delta) \implies h(x) > h(r) - \epsilon > 0$. The problem is that on any interval we can find a number having the form $\displaystyle \tfrac{m}{2^n}$. And $\displaystyle h( \tfrac{m}{2^n} ) = 0 \not > 0$. This is a contradiction.
• Oct 28th 2008, 12:05 PM
rmpatel5
Quote:

Originally Posted by ThePerfectHacker
Let $\displaystyle r\in \mathbb{R}$ with $\displaystyle h(r) > 0$ WLOG. Take $\displaystyle \epsilon > 0$ so that $\displaystyle h(r) - \epsilon > 0$. Then it means there is $\displaystyle \delta > 0$ such that if $\displaystyle x\in (r-\delta,r+\delta) \implies h(x) > h(r) - \epsilon > 0$. The problem is that on any interval we can find a number having the form $\displaystyle \tfrac{m}{2^n}$. And $\displaystyle h( \tfrac{m}{2^n} ) = 0 \not > 0$. This is a contradiction.

would u not have to prove h(r)<0. If so how would i go about doing that??
• Oct 28th 2008, 04:17 PM
ThePerfectHacker
Quote:

Originally Posted by rmpatel5
would u not have to prove h(r)<0. If so how would i go about doing that??

If $\displaystyle h(r) < 0$ choose $\displaystyle \epsilon > 0$ so that $\displaystyle h(r) + \epsilon < 0$. Then there is $\displaystyle \delta > 0$ so that $\displaystyle x\in (r - \delta,r+\delta) \implies h(x) < h(r) + \epsilon < 0$. But since we can find $\displaystyle \tfrac{n}{2^m}$ in this interval it would mean $\displaystyle 0=h(\tfrac{n}{2^m}) < 0$. A contradiction.

Therefore, the only possibility is that $\displaystyle h(r) = 0$ for all $\displaystyle r\in \mathbb{R}$.