# Let $\phi(x)\;andh(x)$ be derivable at x = c. Show that necessary and sufficient condition for the function defined as :$f(x)= \left\{ \begin{array}{1 1} \phi(x), & \quad x \leq c \\ h(x), & \quad x>c \end{array} \right.$to be derivable at x= c are $(i)\phi(c)=h(c)\quad(ii)\phi'(c)=h'(c)$

• Sufficient condition $\phi$ L.H.D and R.H.D should be equal at that point.
$f(x)$ is continuous
$\Rightarrow L.H.L = R.H.L = f(l)$
$\Rightarrow \phi (c)=h(c)$