Is there a function $f:R\to R$ that satisfies
$f$ is everywhere discontinuous.
$\lim_{h \to 0}(f(a+h)-f(a-h))=0$ for all $a \in R$
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1No. See my answer to Does $\lim_{h\rightarrow 0}\ [f(x+h)-f(x-h)]=0$ imply that $f$ is continuous? – Dave L. Renfro Jan 31 '20 at 17:21