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"Let f a function which satisfies $$|f(x)|\leq|x| \forall{x\in{\mathbb{R}}}$$

Proof that is continuous at 0.

We concluded that since f(0)=0 then we found a delta equal epsilon so $$|f(x)|≤|x|<ϵ$$.

But now I have:

$$\textrm{g continuous at 0 and g(0)=0}\Longleftrightarrow{\displaystyle\lim_{x \to{0}}{g(x)}=g(0)=0} $$

$$ \left . \begin{matrix}{g(0)=0}\\{|f(x)|\leq |g(x)|}\end{matrix}\right \} \Longrightarrow{}f(0)=0$$

And I have to prove that f is cont. at 0.

So I have to fid a delta such that:

$$\forall{\varepsilon>0}\textrm{ } \exists{ \delta>0}: \textrm{if } 0<|x-0|<\delta \Longrightarrow{|f(x)-0|< \varepsilon}$$

And I'm in the same problem as before, right? I just choose δ equal ε . If this is OK then I don't understand why it tells me that g is cont. at 0 because I never used that.

Thank you!!