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Find all points of discontinuity of \(f\), where \(f\) is defined by $f(x) = \left\{ \begin{array} {1 1} x + 1,& \quad\text{ if $ x $ \(\geq 1\)}\\ x^2+1,& \quad \text{if $x$ < 1}\\ \end{array} \right. $

$\begin{array}{1 1} \text{The point of discontinuity is} x=2 \\ \text{The point of discontinuity is} x=1 \\ \text{The point of discontinuity is}x=0 \\ \text{There is no point of discontinuity at any point} x\in R \end{array} $

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Toolbox:
  • If $f$ is a real function on a subset of the real numbers and $c$ be a point in the domain of $f$, then $f$ is continuous at $c$ if $\lim\limits_{\large x\to c} f(x) = f(c)$.
Step 1:
At $x=1$
LHL=$\lim\limits_{\large x\to 1^-}f(x)=\lim\limits_{\large x\to 1^-}(x^2+1)$
$\qquad\qquad\quad\quad\;=((-1)^2+1)$
$\qquad\qquad\quad\quad\;=(1+1)$
$\qquad\qquad\quad\quad\;=2$
RHL=$\lim\limits_{\large x\to 1^-}f(x)=\lim\limits_{\large x\to 1^-}(x+1)$
$\qquad\qquad\quad\quad\;=(1+1)$
$\qquad\qquad\quad\quad\;=2$
$f$ is continuous at $x=1$
Step 2:
At $x=c > 1\lim\limits_{\large x\to c}f(x)=\lim\limits_{\large x\to c}(x+1)=c+1=f(c)$
$\Rightarrow f$ is continuous at $x=c > 1$
Step 3:
At $x=c < 1\lim\limits_{\large x\to c}f(x)=\lim\limits_{\large x\to c}(x^2+1)$
$\qquad\qquad\qquad\qquad\;\;=c^2+1$
$\qquad\qquad\qquad\qquad\;\;=f(c)$
$\Rightarrow f$ is continuous at $x=c < 1$
Step 4:
There is no point of discontinuity at any point $x\in R$
answered May 28, 2013 by sreemathi.v
 

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