A function is continuous on a set when it is continuous at every point of it, and that is all the phrase means: the radius may be chosen afresh at each point, and usually must be. Where continuity fails, the failure has a shape, and there are only three of them — a value in the wrong place, a jump, and everything else. Naming which one has occurred is the first thing to do with a discontinuous function.

A road surface is sound if it is sound at every point, and an inspector checks each metre separately with whatever care that metre needs. A fault is either a loose stone that could be replaced, a step between two sound stretches, or something that cannot be described so simply — and the repair bill depends entirely on which.

$f$ is continuous on $A$ when it is continuous at each point of $A$. At a point where continuity fails, exactly one of three things has happened: both one-sided limits exist and are equal, so the value alone is wrong — a removable discontinuity; both exist and differ — a jump; or at least one fails to exist — an essential discontinuity. The classification is complete because it is a case split on the one-sided limits.

• Removable: $\tfrac{x^2-1}{x-1}$ at $1$, or any function whose value has been moved off the limit.
• Jump: at , or any step. A monotone function's only discontinuities are jumps, and it has at most countably many.

Take any point and show that continuity fails there, whichever kind of number it is. Density gives sequences of rationals and of irrationals converging to the point, and along the first the values are always one while along the second they are always zero. Whatever the value at the point, one of the two sequences reports the wrong answer, so the sequential form of continuity fails. No point of the line escapes, because both the rationals and the irrationals reach into every interval.

Take any point of the line whatsoever.

Density of the rationals supplies a sequence of them converging to the point.

Density of the irrationals supplies another.

The two sequences of values head to different numbers.

Continuity would force both to converge to the value at the point, and they cannot both.

Split on whether both one-sided limits exist, and then on whether they agree.

• Both exist and agree: removable.
• Both exist and differ: a jump.
• At least one fails: essential.

• Continuity on a set is continuity at each of its points, with a radius chosen afresh at each.
• Removable, jump and essential exhaust the failures, by a case split on the one-sided limits.
• A monotone function has only jumps.
• A function can be discontinuous at every point of the line.

Great work! You now know how to:

• Say what continuity on a set does and does not promise
• Classify a discontinuity as removable, a jump or essential
• Give a function discontinuous at every point and say why
• Say what monotonicity rules out