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I am having a midterm review in school and there's one concept that I forgot how to solve, and that is solving for continuous functions?

More precisely, what does a variable have to be for the following to be continuous. For example, the problem I am dealt with solving is $$F(x)=\left\{\begin{array}{ll} 2x&\text{if }x\leq 1\\ ax^2+1&\text{if }x\gt 1\\ \end{array}\right.$$ and I have to solve for $a$. Normally, I would solve for $ax^2+1$, but I know that is wrong. Can someone tell me how to solve this, and perhaps by using a different problem so that I may be able to do the one I have on my own?

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    You need to make $F$ continuous when $x = 1$. What is $F(1)$? What must $\lim_{x \rightarrow 1} F(x)$ equal in order for $F$ to be continuous at $x = 1$? To compute the limit, what are the one-sided limits, $\lim_{x \rightarrow 1^-} F(x)$ and $\lim_{x \rightarrow 1^+} F(x)$? Hope those hints help.2012-01-17
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    The only way to acknowledge the help of answerers in this forum is by accepting an answer. So, please consider doing so.2012-01-17
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    Please see [this](http://meta.math.stackexchange.com/q/3286/8271)2012-01-17

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