Q.Find the values of $a$ and $b$ so that $f(x) = \begin{cases} 1, & x \le 0 \\ ax + b, & 0 < x < 1 \\ 5, & x \ge 1 \end{cases}$ is continuous at both $x = 0$ and $x = 1$.

Match the pieces at the two joining points $x=0$ and $x=1$ to get two equations in $a$ and $b$.

Apply the continuity condition at each joining point (§6, Case C).

Continuity at $x = 0$. The left piece is the constant $1$ (valid for $x \le 0$, giving $f(0) = 1$); the middle piece is $ax + b$ (valid for $0 < x < 1$):

For continuity, $b = 1$. …(i)

Continuity at $x = 1$. The middle piece $ax + b$ must match the right constant $5$ (valid for $x \ge 1$, giving $f(1) = 5$):

For continuity, $a + b = 5$. …(ii) …

A common error is to set $b$ equal to the wrong side at $x=0$ (e.g. equating the middle piece to $5$ instead of $1$). Match each joining point to the piece that actually borders it: at $x=0$ the middle meets the left constant $1$; at $x=1$ the middle meets the …