Let $f$ be differentiable function from $\mathbb{R}$ to $\mathbb{R}$. Consider the set $$A_y=\{x \in \mathbb{R} : f(x)=y \}$$ I want to know whether $A_y$ is countable for each $y\in \mathbb{R}$. I can verify using simple function like polynomial , exponential function, sine, cosine function; it is countable there. Is it true for any differentiable function?
$\begingroup$
$\endgroup$
5
-
1$\begingroup$ Take a constant function ;) $\endgroup$– DodoDuQuercyCommented Jul 7, 2020 at 6:50
-
$\begingroup$ Can we consider constant function on all $\mathbb{R}$? $\endgroup$– zkutchCommented Jul 7, 2020 at 6:50
-
$\begingroup$ what if it is non constant $\endgroup$– Madhan KumarCommented Jul 7, 2020 at 6:52
-
$\begingroup$ Of course you can! $\endgroup$– DodoDuQuercyCommented Jul 7, 2020 at 6:52
-
$\begingroup$ What is a differential function ? $\endgroup$– user65203Commented Jul 7, 2020 at 6:55
Add a comment
|
2 Answers
$\begingroup$
$\endgroup$
0
No, for the simple reason that constant functions exist.
But there are also nontrivial examples: the function $$ f(x) = \begin{cases} e^{-1/x^2}, &\text{if } x>0, \\ 0, &\text{if } x\le 0 \end{cases} $$ is differentiable everywhere.
answered Jul 7, 2020 at 6:51
$\begingroup$
$\endgroup$
$$f(x):=y$$ is a differentiable function and $$\{x\in\mathbb R:f(x)=y\}=\mathbb R.$$
You can conclude.
For a less trivial example,
$$f(x):=(x+|x|)^2,y=0.$$