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I have two functions:

  1. $f: \mathbb{R} \to \mathbb{R}: x \mapsto |x|$

  2. $f: \mathbb{R} \to \mathbb{R}: x \mapsto 3x^2-7x^2+11x-1$

I´m not really sure how to approach the question whether these functions are continuous or not. For the 1. because it is not differentiable at $0$ than its not continuous? For the 2. its continuous because its differentiable?

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    $\begingroup$ While what you say in 2. is true, this is not the appropriate argument. Rather, appeal to the facts that $f(x)=x$ is continuous, constant functions are continuous, and sums, products, and constant multiples of continuous functions are continuous. $\endgroup$ – David Mitra Jan 24 '14 at 15:52
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On 2: It is. Every differentiable function is continuous. But not every continuous function is differentiable.

On 1: The only problem is $x=0$. To see it is indeed continuous, you have to look at the limits $x\to0$ for $x<0$ and $x>0$. Both limites are $0$, so the function is continuous.

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Differentiability implies continuity (another proof), but not conversely.

The definition of Continuity is available here

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In a metric space, like $\mathbb R$ (if it is equipped with its usual topology) it is enough to prove or check that $x_n\rightarrow x$ implies that $f(x_n)\rightarrow f(x)$.

Off course this is not a general approach, but it can be applied in many cases.

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