Find all points of discontinuity of f, where

$f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{\cfrac{{\sin x}}{x},}&{if}&{x < 0}\\{x + 1,}&{if}&{x \ge 0}\end{array}} \right.$

At $x = 0,f(0) = 1$

L.H.L.$= \mathop {\lim }\limits_{x \to {0^ - }} f(x) = \mathop {\lim }\limits_{\scriptstyle x \to 0 - h\atop\scriptstyle h \to 0} \cfrac{{\sin ( - h)}}{{ - h}} = 1$

R.H.L.$\mathop {\lim }\limits_{x \to {0^ + }} f(x) = \mathop {\lim }\limits_{\scriptstyle x \to 0 + h\atop\scriptstyle h \to 0} (h + 1) = 0 + 1 = 1$

therefore, $\mathop {\lim }\limits_{x \to {0^ - }} f(x) = \mathop {\lim }\limits_{x \to {0^ + }} f(x) = f(0)$
Thus, f(x) is continuous at x $=$ 0.

When $x < 0$, sin x and x both are continuous.
therefore, $\cfrac{{\sin x}}{x}$ is also continuous.

When $x > 0,$ f(x) $=$ x + 1 is a polynomial.

therefore, f(x) is continuous.

Hence we can say that f(x) is not discontinuous at any point.