Proof of continuity using Epsilon-Delta

Question

Let $$f(x) = \left\{ \begin{array}{lr} x^2 \cos\left(\frac{1}{x}\right) &: x \ne 0 \\ 0 &: x = 0 \end{array}\right.$$ Prove that $f(x)$ is continuous for all $x \in \mathbb{R}$.

I can use a theorem which states: let $I, J \subseteq \mathbb{R}$ be intervals. Let $f:I \to \mathbb{R}$ and $g:J \to \mathbb{R}$ and suppose $f(I) \subseteq J$. Suppose $f$ is continuous at $y \in I$ and $g$ is continuous at $f(y)$. Then $g \circ f$ is continuous at $y$.

score 1 · Answer 1 · answered Oct 24 '13 at 03:30

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$\textbf{Hint:}$ Use your theorem to prove that $f$ is continuous on $\mathbb{R} - \{0\}$, and prove that $f$ is continuous in $0$ with an $\epsilon-\delta$ argument.

answered Oct 24 '13 at 03:30

Arthur

5,524

(In fact, one cannot use the theorem on $x=0$!) – Pedro Oct 24 '13 at 03:33

score 0 · Accepted Answer · answered Oct 24 '13 at 03:37

0

Hint: For the case $x=0$, note that

$$|x^2\cos(1/x)-0|\leq |x|^2 < \epsilon. $$

answered Oct 24 '13 at 03:37

Mhenni Benghorbal

47,431

How so? I'm just not seeing it. – Maddy Oct 24 '13 at 04:04
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Nevermind, I think I got it. Thank you. – Maddy Oct 24 '13 at 04:23
@Maddy: you are welcome. – Mhenni Benghorbal Oct 24 '13 at 16:32

Proof of continuity using Epsilon-Delta

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