Continuity — Question 2

Question 2

For real constants $a$ and $b$, let $$f(x)=\begin{cases} \dfrac{\sin(ax)}{x},&x\ne0,\\[6pt] b,&x=0. \end{cases}$$ Classify the pairs $(a,b)$ for which $f$ is continuous at zero and those for which it has a removable discontinuity there.

State how to remove the discontinuity.

Question 2 - Solution

For $a\ne0$, write

$$\frac{\sin(ax)}x=a\frac{\sin(ax)}{ax}\longrightarrow a.$$

For $a=0$, the quotient is identically zero away from zero, so the same limit holds.

Thus $\lim_{x\to0}f(x)=a$ for every real $a$.

Since $f(0)=b$, continuity holds exactly when $\boxed{b=a}$.

When $\boxed{b\ne a}$ the limit exists but differs from $f(0)$, so the discontinuity is removable by resetting $f(0)=a$.

A function cannot be both continuous and discontinuous at the same point.