Continuity — Question 2

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.