Relationship between connectedness and continuity

You're right.

a) If $f:\mathbb{R}^n\longrightarrow\mathbb{R}$ is continuous, then the graph of $f$ is path connected, whence connected (see here, for instance).

Proof: if $(x,f(x))$ and $(y,f(y))$ are two points on the graph, then $$ t\longmapsto ((1-t)x+ty,f((1-t)x+ty)) $$ is a continuous path connecting them within the graph. QED.

b) Your example is great. This graph is connected and the function is yet discontinuous.

Note: I should add that in general, if $f:C\longrightarrow X$ is continuous, and if $C$ is connected, then the graph of $f$ is connected in $C \times X$ as the continuous image of $C$ under $x\longmapsto (x,f(x))$. This is because continuity preserves connectedness.