Example · Undergraduate
Terms
| Term | This example | Why |
|---|
| limit of function |
$\lim_{x \to 0} f(x)$ does not exist
|
The left and right limits differ.
|
| left limit |
$\lim_{x \to 0^{-}} f(x) = 0$
|
|
| right limit |
$\lim_{x \to 0^{+}} f(x) = 1$
|
The step function jumps at $0$, so its left limit is $0$ and its right limit is $1$.
|
| jump discontinuity |
✓ |
The step function has a jump discontinuity at $0$, because the left limit at $0$ is $0$ and the right limit at $0$ is $1$.
|
| sequentially continuous |
✗ |
$x_n = -1/n \to 0$, but $f(x_n) = 0$ does not approach $f(0) = 1$.
|
| continuous |
✗ |
|
| removable discontinuity |
✗ |
The step function has no removable discontinuity at $0$, because its left and right limits at $0$ differ, so $\lim_{x \to 0} f(x)$ does not exist.
|
| infinite discontinuity |
✗ |
The step function has no infinite discontinuity at $0$, because its values are only $0$ and $1$.
|
| oscillating discontinuity |
✗ |
The step function has no oscillating discontinuity at $0$, because both one-sided limits at $0$ exist.
|
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