A function is continuous at a point when its value there agrees with the value approached by the function from both sides. Formally, is continuous at if .
A function is continuous at a point when its value there agrees with the value approached by the function from both sides. Formally, is continuous at if .
Intuitively, the graph has no hole, jump, or break at . You can pass through that point without lifting your pencil, although this graphical description is only an intuition rather than a complete mathematical definition.
For continuity at , three conditions must all hold:
| Condition | Meaning |
|---|---|
| exists | The function is defined at the point. |
| exists | The left-hand and right-hand limits exist and are equal. |
| The limiting value equals the actual function value. |
The second condition is often decisive. A two-sided limit exists only when
The value of does not affect whether this limit exists; it matters in the third condition. If both one-sided limits equal , but is missing or differs from , there is a removable discontinuity. Defining repairs it. Unequal one-sided limits instead produce a jump discontinuity, which cannot be repaired by changing only .
Consider
At ,
and . Therefore, is continuous at . If instead , the limit would remain , but the function would be discontinuous.
This also explains why changing a single point usually does not change a limit. Limits describe nearby behavior, whereas continuity compares that nearby behavior with the assigned value. On a graph, inspect both sides first, then check the plotted point at .
A common misconception is that continuity follows merely because exists. This is false: all three conditions are necessary. Also, differentiability implies continuity, but continuity does not imply differentiability; is continuous but not differentiable at .
Exam technique: This is Mathematics AA HL, AHL 5.12. The syllabus requires informal understanding, but examinations will not ask you to test continuity and differentiability directly. Apply the concept when interpreting limits, graphs, and derivatives on Paper 1 or Paper 2.