  We give a characterization of zero divisors of the ring $C[a,b].$ Using the
Weierstrass approximation theorem, we completely characterize topological
divisors of zero of the Banach algebra $C[a,b].$ We also characterize the zero
divisors and topological divisors of zero in $\ell^\infty.$ Further, we show
that zero is the only zero divisor in the disk algebra
$\mathscr{A}(\mathbb{D})$ and that the class of singular elements in
$\mathscr{A}(\mathbb{D})$ properly contains the class of topological divisors
of zero. Lastly, we construct a class of topological divisors of zero of
$\mathscr{A}(\mathbb{D})$ which are not zero divisors.
