A proof involving continuity?

**mfetch22** · August 10th 2010, 08:41 AM

Okay, so heres the problem:

Suppose $f$ is the function satisfying $|f(x)| \leq |x|$ for all $x$ . Show that $f$ is continuous at $0$ .

Okay, this is my work so far:

We need to show $\lim_{x \to 0} f(x) = f(0)$ , I started by figuring the value of $f(0)$ . To find that value, I proceeded with:

Since:

$x = 0$

and:

$|f(x)| \leq |x|$

then:

$|f(0)| \leq |0| \;\; \Rightarrow \;\; |f(0)| \leq 0$

but:

$|a| \geq 0 \;\;\;\; for \;\; all \;\; a$

So we must have:

$|f(0)| \geq 0$

But we also have the restriction that:

$|f(0)| \leq 0$

Therefore it must be true that:

$|f(0)| = 0 \;\; \Rightarrow \;\; f(0) = 0$

So, now i just need to show that:

$0 < |x-0| < \delta \;\; \Rightarrow \;\; |f(x) - f(0)| < \epsilon$

but since $f(0) = 0$ ; we need only show:

$0 < |x| < \delta \;\; \Rightarrow \;\; |f(x)| < \epsilon$

This is where i'm stuck. I betting the answer is right under my nose, and is really obivous, and I'm just not seeing it. I appreciate any help that anybody can give.

**Plato** · August 10th 2010, 08:45 AM

Just let $\varepsilon = \delta$

**mfetch22** · August 10th 2010, 08:50 AM

Originally Posted by Plato

Just let $\varepsilon = \delta$

I'm not questioning your answer, I was just wondering if you could show me what about:

$0 < |x-0| < \delta$

and letting:

$\delta = \epsilon$

Implies that $|f(x) - f(0)| = |f(x)| < \epsilon$

??

I'm guessing the answer is probably blatantly obvious or something, but I'm new to these proofs so I'm still learning. Thanks

**Also sprach Zarathustra** · August 10th 2010, 08:51 AM

Prove something stronger:

If $f(x)$ defined on $(a,b)$ , and suppose there exist $L\in\mathbb{R}$ so to every two points $x_1$ and $x_2$ in $(a,b)$ ,

$|f(x_1)-f(x_2)|\leq L|x_1-x_2|$

then $f(x)$ continuous on $(a,b)$

**Plato** · August 10th 2010, 08:53 AM

$\left| {f(x) - f(0)} \right| = \left| {f(x) - 0} \right| = \left| {f(x)} \right| \leqslant \left| x \right| < \delta = \varepsilon$

**Defunkt** · August 10th 2010, 08:56 AM

Originally Posted by mfetch22

I'm not questioning your answer, I was just wondering if you could show me what about:

$0 < |x-0| < \delta$

and letting:

$\delta = \epsilon$

Implies that $|f(x) - f(0)| = |f(x)| < \epsilon$

??

I'm guessing the answer is probably blatantly obvious or something, but I'm new to these proofs so I'm still learning. Thanks

$|f(x)| \le |x| < \delta$

EDIT: Darn you Plato. Beaten again.

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