Determine whether function continuous or not.
\begin{equation} f(x)= \begin{cases} 3x-1&\text{if }x<1\\ 4&\text{if }x=1\\ 2x&\text{if }x>1 \end{cases} ,~~ c=1\end{equation}
Is this function continuous at $x=c$?
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$\begingroup$In order for $f$ to be continuous at $1$, we need to see if $$\lim_{x\to 1}f(x) \quad\text{and}\quad f(1)$$ both exist and are equal.
To do so, compute the limit from the left, the limit from the right, and $f(1)$. If $$ \lim_{x\to1^-}f(x) = f(1) = \lim_{x\to1^+}f(x), $$ then $f$ is continuous at $1$. If one of the equalities doesn't hold, then $f$ is not continuous at $1$.
I'll let you take it from here.
$\endgroup$ 5 $\begingroup$Hint
Continuous means that the function's graph should atleast visually look continuous.
Try plotting it's graph around $x=1$.
Can you solve it now?
Tip
Consider using MathJax next time. It takes only a few seconds.
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