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Answer by PierreCarre for Proving a function isn't bounded
If $f: \mathbb{R}\setminus \{0\} \to \mathbb{R}$ was bounded there would exist a constant $M >0$ such that$$|f(x)| < M, \quad \forall x\ne 0.$$However, this generates a contradiction since...
View ArticleAnswer by Sammy Black for Proving a function isn't bounded
The formulas that you've written actually prove something stronger, namely that$$\lim_{x \to 0}\; \biggl\lvert \frac1x \biggr\rvert = \infty.$$To establish that $f(x) = \frac1x$ is unbounded, we have...
View ArticleProving a function isn't bounded
Let $f(x) = \frac{1}{x}$ be a function to and from the reals. Is $f$ bounded?I want to prove using the definition. Let $M > 0$ be given. Then there exists a $\delta > 0$ such that:If $0<|x-0|...
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