2.1 Definition of Continuity
Notes
A function is continuous at $x = a$ if you can draw its graph through that point without lifting your pencil. Precisely, all three must hold:
- $f(a)$ is defined.
- $\displaystyle \lim_{x \to a} f(x)$ exists.
- $\displaystyle \lim_{x \to a} f(x) = f(a)$.
- If any condition fails, $f$ is discontinuous at $a$.
- Differentiability is not part of the definition.
Practice
Which of these is NOT required for $f$ to be continuous at $x = a$?
Continuity needs only the three limit conditions. A sharp corner, like $|x|$ at 0, is continuous but has no derivative.
Let $f(x) = x^2 + 1$ for $x \neq 3$ and $f(3) = 4$. Is $f$ continuous at $x = 3$?
$\displaystyle \lim_{x \to 3} f(x) = 10$, which is not equal to $f(3) = 4$.
Why is $\displaystyle f(x) = \frac{x^2 - 1}{x - 1}$ not continuous at $x = 1$?
The denominator is 0 at $x = 1$, so the first condition fails. (The limit does exist and equals 2.)
Ready for more? Take the Unit 2 test.