I am trying to calculate
$$ \dim( \{ f\in C^0 ([a,b]) : f_{|[x_{j-1},x_j]} \in \mathcal{P}_k, j = 1,...,m \}) \text{ with }m,k \in \mathbb{N} $$
Is $m(k+1)$ correct?
My thoughts: I have $m$ continuous piecewise functions of $dim(\mathcal{P}_k)=k+1$. And because the resulting function $f$ is only continuous but not differentiable I need $m(k+1)$ elements to store the coefficients of the $m$ piecewise functions of $f$.
There are $m$ subintervals. On each of them you choose a polynomial of degree $k$, which takes $k+1$ coefficients. So you have $m(k+1)$ coefficients in total, but they are not arbitrary: there are some relations to satisfy.
For each $j=1,\dots,m-1$, the continuity at $x_j$ amounts to a linear equation in the coefficients of polynomials on both sides of $x_j$. None of these equations is a consequence of the others, which is easiest to demonstrate by exhibiting a function that fails continuity at one $x_j$ and no other (i.e., a shift of the Heaviside function).
So, the space you are considering has dimension $m(k+1)-(m-1)$.