The function
[tex]f(x)=\begin{cases}cx^2&\text{for }x\le2\\cx-3&\text{for }x>2\end{cases}[/tex]
is piecewise continuous, since both cx ² and cx - 3 are polynomials. f(x) itself is continuous if both pieces meet at the same defined point. In other words, the limits of f(x) as x → 2 from either side have the same value of f (2) = c•2² = 4c.
We have
[tex]\displaystyle\lim_{x\to2^-}f(x)=\lim_{x\to2}cx^2=4c[/tex]
[tex]\displaystyle\lim_{x\to2^+}f(x)=\lim_{x\to2}(cx-3)=2c-3[/tex]
so in order for f to be continuous, we need
4c = 2c - 3 → 2c = -3 → c = -3/2
