Are there continuous functions $f\colon\mathbb{R}^2\to\mathbb{R}$ such that f has no directional derivatives?

Question

Can the non-differentiability of a function $f:R^n \to R$ always be proved by using directional derivative? leads to the question whether there is some (continuous) function $f\colon\mathbb{R}^2\to\mathbb{R}$ such that for all $(x_0,y_0)\in\mathbb{R}^2$ and all $(\alpha,\beta)\in\mathbb{R}^2$ with $\alpha^2+\beta^2=1$ the following limit $$\lim_{t \to 0}\frac{f((x_0,y_0)+t(\alpha,\beta))-f((x_0,y_0))}{t}$$ does not exist.