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Let ${X_0}$ be a compact Hausdorff space, and let ${\mathbf{C}}({X_0})$ be the Banach algebra of all continuous complex-valued functions on ${X_0}$. It is known that, assuming the continuum hypothesis, any nonmaximal, prime ideal ${\mathbf{P}}$ such that $\vert{\mathbf{C}}({X_0})/{\mathbf{P}}\vert = {2^{{\aleph _0}}}$ is the kernel of a discontinuous homomorphism from ${\mathbf{C}}({X_0})$ into some Banach algebra. Here we consider the converse question of which ideals can be the kernels of such a homomorphism. Partial results are obtained in the case where ${X_0}$ is metrizable.