Example · High school
Terms
| Term | This example | Why |
|---|
| differentiable |
✓ |
$x \mapsto \sin x$ is differentiable at every $x$, with derivative $\cos x$.
|
| continuously differentiable |
✓ |
$x \mapsto \sin x$ is continuously differentiable, because its derivative $\cos x$ is continuous.
|
| smooth |
✓ |
$x \mapsto \sin x$ is smooth, because its derivatives $\cos x$, $-\sin x$, $-\cos x$, $\sin x$, … repeat and all exist.
|
| analytic |
✓ |
$\sin$ is analytic because its Taylor series converges to it everywhere.
|
| polynomial function |
✗ |
$\sin x$ is not a polynomial function. It has infinitely many roots, but a nonzero polynomial has only finitely many.
|
| convex function |
✗ |
The function $\sin$ is not convex, because its second derivative, $-\sin x$, is negative on $(0, \pi)$.
|
Distinctions using this example
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