# Solution: 2011-23 Constant Function

Let $$f:\mathbb{R}^n\to \mathbb{R}^{n-1}$$ be a function such that for each point a in $$\mathbb{R}^n$$, the limit $$\lim_{x\to a} \frac{|f(x)-f(a)|}{|x-a|}$$ exists. Prove that f is a constant function.

The best solution was submitted by Minjae Park (박민재), 2011학번. Congratulations!

Here is his Solution of Problem 2011-23.

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Let X be the set of all postive real numbers c such that  $\frac{\prod_{k=1}^{n-1} \sin\left( \frac{k \pi}{2n}\right)}{c^n}$  converges as n goes to infinity. Find the infimum of X.