The difference function \(\Delta_hf\) is defined by \(\Delta_hf(x)=f(x+h)-f(x)\). The author considers problems of the following type: Let \(\mathcal G\subset\mathcal F\) be classes of periodic functions. For which sets of reals \(H\) it is true that if \(f\in\mathcal F\) and \(\Delta_hf\in\mathcal G\) for every \(h\in H\), then \(f\in\mathcal G\). The class of counter examples is denoted by \(\mathfrak H(\mathcal F,\mathcal G)\). He considers classes of measurable functions on the circle group \(\mathbb T=\mathbb R/\mathbb Z\) closed for changes on null sets, in particular, the class of measurable functions, the classes \(L_p\) and \(L_\infty\), the class of essentially continuous functions, the class of functions with absolutely convergent Fourier series, and the class of essentially Lipschitz functions. Some of the considered classes of functions are shown to have the difference property or the weak difference property. It is observed that the classes \(\mathfrak H(\mathcal F,\mathcal G)\) are often related to some classes of thin sets in harmonic analysis.