Continuity at a point is the limit being what it ought to be. Three things are demanded at once: the function is defined at the point, the limit there exists, and the two agree. The definition is the limit definition with one phrase struck out — the clause excluding the point — and striking it out is exactly what forces the value to join in.

A timetable is continuous at a station if the arrival time listed there is the time the approaching trains are converging on. A listed time that no approach agrees with is a misprint, even though the trains and the list are each perfectly sensible on their own.

The dots trace $f(x)=x$ except at $x=1$, where the value has been moved up. The limit at $1$ is still $1$, because a limit never looks at the point; continuity fails because the value is not the limit.

$f$ is continuous at $c$ when for every $\varepsilon>0$ there is a $\delta>0$ such that $\mid x-c\mid$ implies for every in the domain. This is the limit definition with weakened to and replaced by . Equivalently: is in the domain, exists, and it equals .

• The clause $0<\lvert x-c\rvert$ is gone, so $x=c$ is now tested — and it passes trivially, which is why nothing is lost and the value is forced to agree.
• Sequentially: $f$ is continuous at $$ exactly when implies , with no condition that the inputs avoid .

Compare the two conditions phrase by phrase. The continuity condition tests every nearby input including the point; the limit condition tests every nearby input except the point. The only difference is therefore what happens at the point itself, and there the continuity condition asks that the distance from the value to itself be below the tolerance, which it is. So the two conditions impose exactly the same demands.

Continuity tests more inputs, so it implies the condition on fewer.

Backwards, the one extra input the limit condition does not test passes automatically.

So the same radius serves, and the two conditions are equivalent.

Which is the three-part form the definition is usually quoted in.

The point must be in the domain, the limit must exist, and they must agree.

A continuous function may be taken inside a limit; that is what the sequential form says.

• Continuity at a point asks that the limit exist and equal the value.
• The definition is the limit definition with the exclusion of the point removed.
• Sequentially, a continuous function may be taken inside a limit.
• It is a local condition, and it holds automatically at an isolated point.

Great work! You now know how to:

• State continuity at a point in both the tolerance and the three-part form
• Say which clause of the limit definition is dropped, and why nothing is lost
• Repair a removable discontinuity by assigning the limit
• Move a continuous function through a limit of a sequence