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Find all points of discontinuity of \(f\), where $ f(x) = \left\{ \begin{array} {1 1} \large\frac {sinx} {x},\normalsize & \quad\text{ if \(x\) < 0 }\\ x + 1 ,& \quad \text{if $x$ \( \geq 0\)}\\ \end{array} \right. $

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  • 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 $x=0$
LHL=$\lim\limits_{\large x\to 0}f(x)=\lim\limits_{\large x\to 0}\large\frac{\sin(-h)}{-h}$=1
Therefore $f(0)=1$
RHL=$\lim\limits_{\large x\to 0}f(x)=\lim\limits_{\large x\to 0^+}f(x)$
$\qquad\qquad\quad\;\;=0+1$
$\qquad\qquad\quad\;\;=1$
LHL=RHL=f(0).
$f$ is continuous at $x=0$
Step 2:
When $x<0$ $\sin x$ and $x$ both are continuous.
Therefore $\large\frac{\sin x}{x}$ is also continuous.
When $x>0$ $f(x)=x+1$ is a polynomial $f$ is continuous.
$\Rightarrow f$ is not discontinuous at any point.
answered May 29, 2013 by sreemathi.v
 

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