Parameters for continuity

Problem

Determine $a,b$ such that $f(x)=\begin{cases} ax+1 & x\le 0 \\ b\sin x & 0<x<\pi/2 \\ 2 & x\ge\pi/2\end{cases}$ is continuous on $\mathbb{R}$.

Solution

We impose continuity at the two junction points $x=0$ and $x=\pi/2$.

Junction at $x=\pi/2$: from the left $\lim_{x\to(\pi/2)^-} b\sin x = b\sin\tfrac{\pi}{2}=b$; the right piece equals $2$. Continuity $\iff b=2$.

Junction at $x=0$: the left piece gives $f(0)=a\cdot 0+1=1$; from the right $\lim_{x\to 0^+} b\sin x = b\cdot 0 = 0$. Continuity would require $1=0$, impossible for every value of $a$: the value at $0$ is always $1$, while the sinusoidal piece starts from $0$.

Conclusion: with $b=2$ the function is continuous at $\pi/2$, but at $x=0$ a jump of height $1$ remains: there is no pair $(a,b)$ that makes $f$ continuous on all of $\mathbb{R}$. The parameter $a$ only governs the slope of the first piece, not its value at $0$.

Links

Topics: Functions and properties
Concepts: Continuity · Piecewise function
Skills: Reasoning by cases · Solving equations
Exercise types: Study of a function