Exercise 4.3.9

Question

Assume $h: \mathbf{R} 	o \mathbf{R}$ is continuous on $\mathbf{R}$ and let $K=\{x$ : $h(x)=0\}$. Show that $K$ is a closed set.

Ulisse Mini · Accepted Answer

Let $(x_n)$ be a convergent sequence in $K$ and set $x = \lim x_n$. Since $h$ is continuous the limit $\lim h(x_n) = h(x)$ and since $h(x_n) = 0$ for all $n$ $\lim h(x_n) = 0$. therefore $h(x) = 0$ and $x \in K$.

Exercise 4.3.9

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Exercise 4.3.9

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