Consider the 1-periodic function $f$ with Fourier series $$\sum n^{-1}\cos (2\pi n! x).$$
Note that it satisfies your property, since all but finitely many summands are $h$-periodic for any rational $h$. On the other hand, if it were a Fourier series of a continuous function $F$, its partial sums would be Cesàro convergent to the values of $F$, but for $x=0$ the sum of series is infinite (both Cesàro or usual).

Well, strictly speaking the above function $f$ is defined only on a set of full measure (not pointwise) and $f(x+h)-f(x)$ is equivalent to a continuous function, but not genuine pointwise continuous. Is it ok for you? If not, you may carefully correct it on a set of measure 0. For example, define $f(x)$ as a sum of above series when it converges (by Carleson's theorem the series converges almost everywhere to the initial function). After that it remains to define the values of $f$ on the exceptional set. This exceptional set has measure 0 and is invariant under shifts by rational numbers, we must force the difference $f(x+h)-f(x)$ take the necessary values $f_N(x+h)-f_N(x)$, where $N$ is chosen so that $N!h$ is integer, and $f_N$ denotes the corresponding finite sum. This is possible since these values agree in a natural sense.

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