I have the function $f:\mathbb{R}\rightarrow \mathbb{R}; f(x) = \sqrt{x^2+x+1}$. I don't know exactly what should I do since the squre root function is defined only for positive real numbers, and f takes inputs from the whole set of real number. Also if I would study the continuity for $x^2+x+1$ I know that this function is continuous everywhere since it is a polynomial.
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what is the question here? – Dr. Sonnhard Graubner Jan 21 '17 at 14:26
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Did you ask yourself how can $x^2 + x + 1$ be negative? – Daniel Jan 21 '17 at 14:26
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I need to tell if the function is continuous or not (and why ) – Raducu Mihai Jan 21 '17 at 14:27
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it is clear that $$p(x)=x^2+x+1$$ is continuously since $$p(x)$$ is a polynomial – Dr. Sonnhard Graubner Jan 21 '17 at 14:28
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Yeah but my function is $\sqrt{p(x)}$ – Raducu Mihai Jan 21 '17 at 14:30
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$x^2 + x +1 = 0 $ has no solution. As the term with the highest degree is positive, the function is always positive. Thus the square root function gets only positive input for $\forall x \in \mathbb{R}$, and so the function $f(x)$ is continuous on the whole $\mathbb{R}$.
user401855
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Thanks that clears the things a bit. But I should give a $\epsilon - \delta$ argument. – Raducu Mihai Jan 21 '17 at 14:33