(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that the function:

[tex]\frac{2x-1}{x^2+1}, x \in \mathbb{R}[/tex]

is continuous.

2. Relevant equations

Definition 1.

The function y=f(x) satisfied by the set D_{f}is continuous in the point x=a only if:

1^{0}f(x) is defined in the point x=a i.e. [itex]a \in D_f[/itex]

2^{0}there is bound [tex]\lim_{x \rightarrow a}f(x)[/tex]

3^{0}[tex]\lim_{x \rightarrow a}f(x)=f(a)[/tex]

Theorem 1.

If the functions y=f(x) and y=g(x) are continuous in the point x=a Є D_{f}∩ D_{g}, then in the point x=a these functions are continuous:

y=f(x)+g(x), y=f(x)g(x) and y=f(x)/g(x), if g(a) ≠ 0.

3. The attempt at a solution

I tried using the definition 1.

But also this function is composition of two functions f(x) and g(x), so can I use the fact that f(x)=2x-1 and g(x)=x^{2}+1 are continuous, and y=f(x)/g(x), g(a) ≠ 0 since x^{2}+1 ≠ 0 ?

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# Homework Help: [prove] Continuous function

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