Continuity at $x=1$: $\lim_{x\to1^+}(a+bx)=a+b=5$ ...(i)

Continuity at $x=3$: $a+3b=b+15 \Rightarrow a+2b=15$ ...(ii)

Continuity at $x=5$: $b+25=30\Rightarrow b=5$ ...(iii)

From (iii): $b=5$. From (i): $a=5-5=0$... but check (ii): $0+10=10\neq15$. Contradiction!

Try option B: $a=-5, b=10$. Check (i): $-5+10=5$ ✓. Check (ii): $-5+20=15$ ✓. Check (iii): $10+25=35\neq30$ ✗.

No values make $f$ continuous, so the answer is not continuous for any $a, b$ (option D, index 3).