What continuity at a point really requires

The three-condition checklist

First, f(a) must exist. Second, the left- and right-hand limits must agree. Third, the shared limit must equal f(a). Missing any one condition produces a discontinuity.

This separates holes, jumps, and vertical blowups instead of calling every visual break the same thing.

Continuity is local

A function can be continuous at one point and fail nearby. The definition makes a claim only about inputs sufficiently close to a and the value at a itself.

For familiar elementary functions, continuity lets you evaluate limits by substitution wherever the expression is defined.

Repairing a removable discontinuity

If the limit exists but the function value is missing or wrong, assigning the limiting value repairs the point. This changes one output without changing the nearby rule.

Worked example

Common mistakes

• Checking only that f(a) exists.
• Assuming a graph is continuous because the pieces touch visually.
• Using a two-sided limit when the domain naturally ends at the point.

Keep these ideas

• Defined, limit exists, and values match.
• Continuity turns many limits into substitution.
• A removable hole can be repaired with one value.