# General approach to find continuous functions?

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?

• 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. – David Mitra Jan 24 '14 at 15:52

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.
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 $f(x_n)\rightarrow f(x)$.