 Solution

The function $f$ is differentiable at $c$ if the finite limit of a functionexists. If it exists, put $L=f^{\prime }(c)$ and defineThe definition of derivative gives $ϵ(x)\to 0$ as $x\to c$. For $x\ne c$, rearrangement gives the desired local linear approximationAt $x=c$, this identity holds because both terms containing $x-c$ vanish. Conversely, if such an identity holds, then for $x\ne c$ its difference quotient is $L+ϵ(x)\to L$. Hence $f$ is differentiable at $c$, and necessarily $\boxed{L=f^{\prime }(c)}$.