We study the de Rham function: the unique continuous (nowhere differentiable) function $ F \in L^1(\mathbb{R})$ with $ \int F(x)\,dx=1$ satisfying the functional equation $ F(x)=F(3x)+\frac{1}{3}\bigl(F(3x-1)+F(3x+1)\bigr)+\frac{2}{3}\bigl(F(3x-2)+F(3x+2)\bigr)$. We show that its pointwise Hölder regularity $ \alpha(x)=\liminf_{h\to 0}\frac{\log(\vert F(x+h)-F(x)\vert)}{\log \vert h\vert}$ differs widely from point to point, and the values of $ \alpha(x)$ fill an interval parametrizing the fractal sets $ E^{(\alpha)}$, where $ E^{(\alpha)}$ is the set of points $ x$ with Hölder exponent $ \alpha(x)=\alpha$. We also prove that the thermodynamic formalism (increment method) holds for the de Rham function: we have a heuristic formula $ d(\alpha)=\inf_{q >0}(\alpha q-\zeta(q)+1)$ relating the order of decay of $ \int_{\mathbb{R}}\vert F(x+h)-F(x)\vert^{q}\,dx\sim \vert h\vert^{\zeta(q)}$ as $ h \to 0$ with the Hausdorff dimension $ d(\alpha)$ of $ E^{(\alpha)}$.