Defined
The function value f(c) exists.
Function value · two-sided limit · local agreement
Continuity
Continuity at a point is a three-way agreement: the function must be defined there, nearby values must approach one two-sided limit, and that limit must equal the actual function value. It is stronger than simply having a limit.
Point continuity
The limit describes nearby behavior; continuity additionally anchors that behavior to the point itself.
Limit exists
The left-hand and right-hand limits agree on one finite value L.
Value matches limit
The actual value satisfies f(c) = L.
Removable discontinuity lab
Keep the nearby curve fixed. Change only f(2).
For x ≠ 2, use f(x) = (x−2)² + 3. Nearby values approach 3 from both sides, so the two-sided limit is fixed at 3. Continuity depends on whether the actual point value agrees with that limit.
removable discontinuity
Set f(2)
f(2) = 1.50
Classifying discontinuities
Ask which part of the continuity agreement fails.
The labels describe local behavior near a point. A function can have different discontinuity types at different inputs.
Removable
The two-sided limit exists, but the function is missing or assigned a different value at the point. Redefining one point can repair continuity.
Jump
The left-hand and right-hand limits approach different finite values, so there is no single two-sided limit to match.
Infinite
At least one one-sided limit is unbounded near the point, producing vertical-asymptote behavior rather than a finite continuous connection.
Oscillatory
Nearby outputs do not settle toward one value because the function continues oscillating at every scale near the point.
Continuity on an interval
Pointwise continuity builds interval-level guarantees.
If a function is continuous at every point of an interval, the graph cannot jump over intermediate output values. This is the idea behind the Intermediate Value Theorem: on a closed interval, any value between f(a) and f(b) is attained somewhere between a and b.