Explanation

(c)
Continuity at $\mathbf{x = 0}$
LHL=$\lim_{x \rightarrow 0^{-}}\frac{\tan x}{x} = \lim_{h \rightarrow 0}\frac{- tan\ h\ }{- h} = 1$
RHL$= \lim_{x \rightarrow 0^{+}}\frac{\tan x}{x} = \lim_{h \rightarrow 0}\frac{tan\ h\ }{h} = 1$
$\therefore$ LHL=RHL$= f(0) = 1$, it is continuous
Differentiability at $\mathbf{x = 0}$
LHD$= \lim_{h \rightarrow 0}\frac{f(0 - h) - f(0)}{- h} = \lim_{h \rightarrow 0}\frac{\frac{\tan( - h)}{- h} - 1\ }{- h}$
$= \lim_{h \rightarrow 0}\frac{+ \frac{h^{2}}{3} + \frac{2h^{4}}{15} + \ldots}{- h} = 0$
RHD$= \lim_{h \rightarrow 0}\frac{f(0 + h) - f(0)}{h} = \lim_{h \rightarrow 0}\frac{\frac{\tan h}{h} - 1}{h}\$
$= \lim_{h \rightarrow 0}\frac{\frac{h^{2}}{3} + \frac{2h^{4}}{15} + \ldots}{- h} = 0$
$\therefore$ LHD=RHD
Hence, it is differentiable.