Example · Undergraduate
Terms
| Term | This example | Why |
|---|
| removable discontinuity |
✓ |
$f(x) = \frac{\sin x}{x}$ for $x \neq 0$, $f(0) = 0$, has a removable discontinuity at $0$, because $\lim_{x \to 0} \frac{\sin x}{x} = 1$ exists but $f(0) = 0 \neq 1$; redefining $f(0) = 1$ makes $f$ continuous.
|
| jump discontinuity |
✗ |
$f(x) = \frac{\sin x}{x}$ for $x \neq 0$, $f(0) = 0$, has no jump discontinuity at $0$, because the left and right limits at $0$ are both $1$.
|
| infinite discontinuity |
✗ |
$f(x) = \frac{\sin x}{x}$ for $x \neq 0$, $f(0) = 0$, has no infinite discontinuity at $0$, because $\abs{f(x)} \leq 1$ for every $x$.
|
| oscillating discontinuity |
✗ |
$f(x) = \frac{\sin x}{x}$ for $x \neq 0$, $f(0) = 0$, has no oscillating discontinuity at $0$, because $f(x)$ tends to $1$ from both sides.
|
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