How do you know if a function is continuous

Continuity means the graph can be drawn without lifting the pencil. Formally, a function f is continuous at a point c if three conditions hold: f(c) is defined, the limit of f(x) as x\to c exists, and that limit equals f(c). If any condition fails, a jump, hole, or asymptote appears.

ε‑δ Definition of Continuity

The rigorous ε‑δ definition states: for every ε>0 there exists a δ>0 such that whenever 0<|x‑c|<δ, we have |f(x)‑f(c)|<ε. This captures the idea that f(x) can be made arbitrarily close to f(c) by restricting x to lie sufficiently close to c. In practice, proving continuity with ε‑δ is rarely required for elementary problems, but the definition underlies all other tests.

Three practical ways to test continuity at a point:

• Check that the function is defined at the point.
• Compute the left‑hand and right‑hand limits and verify they are equal.
• Confirm that the common limit equals the function’s value.

Procedure to verify continuity at c:

1. 1Write down f(c).
2. 2Find $\lim_{x\to c-}f(x)$ and $\lim_{x\to c+}f(x)$.
3. 3If the two limits exist and are equal, call the common value L.
4. 4Check that L = f(c). If all hold, f is continuous at c.

Consider $f(x)=\frac{x^{2}-4}{x-2}$ and test continuity at $c=2$. Direct substitution gives 0/0, so simplify: $f(x)=\frac{(x-2)(x+2)}{x-2}=x+2$ for $x\neq2$. The limit as x\to2 of the simplified expression is $2+2=4$. Since the original definition leaves f(2) undefined, define $f(2)=4$. Then $\lim_{x\to2}f(x)=4=f(2)$, so the function becomes continuous at 2 after the removable discontinuity is filled.

Values used in the example:

If a function is built from continuous pieces—polynomials, exponentials, trigonometric functions, and rational functions with non‑zero denominators—it is automatically continuous on any interval where those pieces apply. To show a function is continuous on an entire interval, verify continuity at each endpoint and use the fact that compositions and sums of continuous functions remain continuous.