Defining continuity at a point

Intuition and formal definition

A function f is continuous at a point a in its domain if its value at a coincides with its limit as x approaches a, without any jumps or gaps. Formally, we define:

f is continuous at a <=> lim(x->a) f(x) = f(a)

In the epsilon-delta version (the rigorous definition to be learnt at a higher level):

For any epsilon > 0, there exists a delta > 0 such that, for any x, if |x - a| ≤ delta, then |f(x) - f(a)| ≤ epsilon.

Detailed example

Let f(x) = x². At a = 3, we have f(3) = 9. We verify that lim(x→3) x² = 9, so f is continuous at 3. Conversely, the floor function f(x) = floor(x) is not continuous at a = 2: the left-hand limit is 1, the right-hand limit is 2; they differ and do not both equal f(2) = 2.

Left-hand and right-hand continuity

Concept	Condition
Left-hand continuity at a	lim(x → a⁻) f(x) = f(a)
Right-hand continuity at a	lim(x → a⁺) f(x) = f(a)
Continuity at a	both of the above are true

Common pitfall

Do not confuse ‘f has a limit at a’ with ‘f is continuous at a’. A function may have a finite limit at a without this limit being equal to f(a) (a point of discontinuity known as ‘removable’). Example: f(x) = (x² - 1)/(x - 1) for x ≠ 1 and f(1) = 5. The limit at 1 is 2, but f(1) = 5, so f is not continuous at 1, even though the limit exists.