# Finding value(s) of a for which f is continuous

A function $f$ is defined as follows: $$f(x)=\begin{cases}\sin x&\text{if }\;x\leq c,\\ax+b&\text{if }\;x>c,\end{cases}$$ where $a,b,c$ are constants. If $b$ and $c$ are given, find all values of $a$ (if any exist) for which $f$ is continuous at the point $x=c$.

So, I'm pretty new to thinking about continuity. I know that I need to show that $$f(c)=\lim_{x\to c}f(x)$$ (and that both exist). However, I'm a little lost. I see that $f(c)=\sin c$, and I think that I've got to find the point where the two cases meet, which should be the limit. $$\sin c=ac+b\implies a=\frac{\sin c-b}{c}.$$ At this point, though, I'm not sure I'm on the right track, and I'd appreciate your help.

• That was really all you had to do. Commented Nov 3, 2013 at 8:27
• You haven't considered the case $c=0$. What if, for example, $b\neq0$? Commented Nov 3, 2013 at 13:16

Therefore $$\lim_{x\to c^-}f(x)=\lim_{x\to c^+}f(x)=f(c)$$ is the condition of for continuity.
In your problem $$\lim_{x\to c^-}f(x)=\lim_{x\to c^-}\sin x=\sin c$$ $$\lim_{x\to c^+}f(x)=\lim_{x\to c^+}ax+b=ac+b$$ and the value of the fucntion at x=c is $\sin c$ . Hence for the function to be continuous you must have $$ac+b=\sin c$$