Condition for $f(x)=p[x+1]+q[x-1]$ to be continuous at $x=1$

Question

The function $f(x)=p[x+1]+q[x-1]$ where $[x]$ is the greatest integer function is continuous at $x=1$, if ___________________

• $f(1)=2p$
• $\lim_{x\to 1^+}f(x)=p(2)+q(0)=2p$
• $\lim_{x\to 1^-}f(x)=p(1)+q(-1)=p-q$
• $p-q=2p\implies p+q=0$

But my reference gives the solution $p=q$, what is going wrong here ?

Answer 1

I think you are right and the textbook is wrong. Try plotting what the textbook says the answer is $p=q$. The textbook's answer produces a stair case. However, your answer, $p=-q$, is a flat line. In terms of wave superposition your answer has a "destructive interference", where the stair cases cancel and this results in a flat line.