Topology and Boundedness

Question

Good people! I am reading general topology in the analysis, and working on a problem: Let $f:[a,b]\to \mathbb R$ be a function such that for every $x \in [a,b]$ there is a $\delta_x>0$ such that $f$ is bounded on $(x-\delta_x, x+\delta_x)$. Prove that $f$ is bounded. How to solve that?Could you guys please help me with that? Thank you in advance.

Answer 1

By the Heine-Borel theorem, there is a finite set $\{x_1,x_2,\ldots,x_n\}\subset[0,1]$ such that$$[0,1]\subset\bigcup_{j=1}^n(x_j-\delta_{x_j},x_j+\delta_{x_j}).$$Now, use the fact that $f$ is bounded on each $(x-\delta_x,x+\delta_x)$.

Answer 2

Cover $[a,b]$ by the open cover $(x-\delta_x, x+\delta_x): x \in [a,b]$ and apply compactness.

Answer 3

Hint: For each $x$, we can find an open ball $U_{x}=(x-\delta_{x},x+\delta_{x})$ on which $|f|\leq M_{x}$ for some positive constant $M_x$. Now, note that $\{U_{x}\}_{x\in[a,b]}$ is an open cover of $[a,b]$. Since $[a,b]$ is compact, we can find a finite subset $\{U_{x_{i}}\}_{i=1}^{n}$ which also covers $[a,b]$. Then...