Q.Find the values of $a$ and $b$ so that $f(x) = \begin{cases} 5x - 2, & x \le 1 \\ ax + b, & 1 < x < 3 \\ 7, & x \ge 3 \end{cases}$ is continuous at both $x = 1$ and $x = 3$.

Continuity at the two joining points gives two linear equations in $a$ and $b$, solved simultaneously (§6, Case C).

The function joins at $x = 1$ and at $x = 3$; continuity at each joining point gives one equation (§6, Case C).

Continuity at $x = 1$. Here the left piece $5x - 2$ (valid for $x \le 1$, so it also gives $f(1)$) must match the middle piece $ax + b$ (valid for $1 < x < 3$):

For continuity, $a + b = 3$. …(i)

Continuity at $x = 3$. Here the middle piece $ax + b$ must match the right piece $7$ (valid for $x \ge 3$, so $f(3) = 7$):

For continuity, $3a + b = 7$. …(ii) …

A frequent error is applying the continuity condition at only one joining point, giving a single equation that cannot pin down two unknowns. With two unknowns $a$ and $b$, two equations are needed — one from each joining point — and they must be solved simultaneously. Another slip is mixing …