Difference equation with log

Question

Can we find continuous $f(x)$ and $d$ such that $$ f(x+1)-f(x) = -\log( c|x| + d ) $$ for all $x$? The constant $c>0$ is specified.

Is $x$ an arbitrary real number? Does $f$ have to be continuous? — 5xum, Dec 02 '14 at 14:07
@5xum He stated that $f$ is continuous, and said for all $x$. — Alice Ryhl, Dec 02 '14 at 15:13
@KristofferRyhl He sure did. After I posted the comment. Also, for all $x$ may still mean "for all $x\in\mathbb R$". — 5xum, Dec 02 '14 at 20:32

score 1 · Answer 1 · answered Jan 18 '15 at 11:48

$f(x+1)-f(x)=-\log(c|x|+d)$

$f(x)=\Theta(x)-\sum\limits_x\log(c|x|+d)$ , where $\Theta(x)$ is an arbitrary periodic function with unit period

$f(x)=\Theta(x)-\log\prod\limits_x(c|x|+d)$ , where $\Theta(x)$ is an arbitrary periodic function with unit period

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