What is the condition of curvature continuity?
Do we only consider the value of the curvature $\kappa$, or do we have to also take account of the direction of the curvature vector $\mathbf{k}$ as well?
Let's say that we have 3 curves: $C(t)$,$B^{(1)}(t)$ and $B^{(2)}(t)$, with the condition that $C(t)$ is connecting with both $B^{(1)}(t)$ and $B^{(2)}(t)$ in $G^0$ and $G^1$ continuity as shown in the figure below. Both $B^{(1)}(t)$ and $B^{(2)}(t)$ have the same curvature $\kappa=a$ at the connecting point. However if we are looking in the context of signed curvature $\kappa_s$, one has a negative sign and another one does not.
$$\kappa=|\kappa_s|$$
Then do we consider both curves $B^{(1)}(t)$ and $B^{(2)}(t)$, as connecting with $C(t)$ in curvature continuity?
A curve embedded in ${\mathbb R}^d$ has $d-1$ "curvatures", all of them positive but the last one, which has a sign. For more details on this see Wilhelm Klingenberg, A course in differential geometry, Springer 1978.
For a curve in ${\mathbb R}^3$ we have the curvature $\kappa>0$ and the torsion $\tau$ which can be positive or negative, depending on whether the curve makes a "positive" or a "negative" helix.
A curve $\gamma$ in ${\mathbb R}^2$ has only one curvature $\kappa$, which can be positive or negative. It is positive when the tangent vector rotates counterclockwise in the direction of increasing $t$ or $s$, and is negative when the tangent vector rotates clockwise. This means that the positivity of $\kappa$ is tied to the conventional "positive rotation sense" in the euclidean plane. The formula for the curvature in terms of the arclength parameter $s$ is $$\kappa(s)=\dot x(s)\ddot y(s)-\ddot x(s)\dot y(s)$$ and for a graph curve $t\mapsto\bigl(t,f(t)\bigr)$ it is $$\kappa(t)={f''(t)\over(1+f'^2(t))^{3/2}}\ .$$ In your figure only the red curve can continuously extend the curvature of the cosine curve at the point $(\pi,-1)$; but the positive curvature values should coincide.