Given : $f(x)$ is differentiable at c. Every differentiable function is continuous.
Hence $f(x)$ is continuous at $x=c.$
\[\lim_{x \to c^{-}} f(x) = \lim_{x \to c^+} f(x) = f(c)\]
\[ \lim_{x \to c} x^2 = \lim_{x \to c} ax+b = c^2 \]
$\implies c^2 =ac+b -----(1)$
$f(x)$ is differentiable at $x=c$
LHD at $x=c $ = RHD at $x=c$
\[ \lim_{x \to c^{-}} \frac{f(x)-f(c)}{x-c} = \lim_{x \to c^{+}} \frac{f(x)-f(c)}{x-c}\]
\[ \lim_{ x \to c} \frac{x^2 - c^2}{x-c} = \lim_{x \to c} \frac{(ax+b) - c^2}{x-c} \]
\[ \lim_{x \to c} x+c = \lim_{x \to c } \frac{a(x-c)}{x-c} \]
\[ \lim_{x \to c} (x+c) = \lim_{x \to c} a \]
$\implies 2c = a -------(2)$
from (1) and (2) we get
$c^2 = 2c^2 +b$
$\implies b = -c^2$
$\therefore a = 2c$ and $b = -c^2$