About shape function for pyramid:
In order to be coherent with the facial elements:
Base: $$ \xi\in[-1,1],\eta\in[-1,1],\zeta=0 $$ Axis: $$ \xi=0,\eta=0,\zeta=[0,1] $$
Linear Pyramid5: $$ \left(\begin{array}{c} -\frac{\left(\eta +\zeta -1\right)\,\left(\xi +\zeta -1\right)}{4\,\left(\zeta -1\right)}\ \frac{\left(\eta +\zeta -1\right)\,\left(\xi -\zeta +1\right)}{4\,\left(\zeta -1\right)}\ -\frac{\left(\eta -\zeta +1\right)\,\left(\xi -\zeta +1\right)}{4\,\left(\zeta -1\right)}\ \frac{\left(\xi +\zeta -1\right)\,\left(\eta -\zeta +1\right)}{4\,\left(\zeta -1\right)}\ \zeta \end{array}\right) $$
Quad Pyramid14: $$ \left(\begin{array}{c} \frac{\eta \,\xi \,\left(2\,\zeta -1\right)\,\left(2\,\zeta -2\right)\,\left(\eta +\zeta -1\right)\,\left(\xi +\zeta -1\right)}{8\,{\left(\zeta -1\right)}^4}\ \frac{\eta \,\xi \,\left(2\,\zeta -1\right)\,\left(2\,\zeta -2\right)\,\left(\eta +\zeta -1\right)\,\left(\xi -\zeta +1\right)}{8\,{\left(\zeta -1\right)}^4}\ \frac{\eta \,\xi \,\left(2\,\zeta -1\right)\,\left(2\,\zeta -2\right)\,\left(\eta -\zeta +1\right)\,\left(\xi -\zeta +1\right)}{8\,{\left(\zeta -1\right)}^4}\ \frac{\eta \,\xi \,\left(2\,\zeta -1\right)\,\left(2\,\zeta -2\right)\,\left(\xi +\zeta -1\right)\,\left(\eta -\zeta +1\right)}{8\,{\left(\zeta -1\right)}^4}\ \zeta \,\left(2\,\zeta -1\right)\ -\frac{\eta \,\left(2\,\zeta -1\right)\,\left(2\,\zeta -2\right)\,\left(\eta +\zeta -1\right)\,\left(\xi +\zeta -1\right)\,\left(\xi -\zeta +1\right)}{4\,{\left(\zeta -1\right)}^4}\ -\frac{\xi \,\left(2\,\zeta -1\right)\,\left(2\,\zeta -2\right)\,\left(\eta +\zeta -1\right)\,\left(\eta -\zeta +1\right)\,\left(\xi -\zeta +1\right)}{4\,{\left(\zeta -1\right)}^4}\ -\frac{\eta \,\left(2\,\zeta -1\right)\,\left(2\,\zeta -2\right)\,\left(\xi +\zeta -1\right)\,\left(\eta -\zeta +1\right)\,\left(\xi -\zeta +1\right)}{4\,{\left(\zeta -1\right)}^4}\ -\frac{\xi \,\left(2\,\zeta -1\right)\,\left(2\,\zeta -2\right)\,\left(\eta +\zeta -1\right)\,\left(\xi +\zeta -1\right)\,\left(\eta -\zeta +1\right)}{4\,{\left(\zeta -1\right)}^4}\ -\frac{\zeta \,\left(2\,\zeta -2\right)\,\left(\eta +\zeta -1\right)\,\left(\xi +\zeta -1\right)}{2\,{\left(\zeta -1\right)}^2}\ \frac{\zeta \,\left(2\,\zeta -2\right)\,\left(\eta +\zeta -1\right)\,\left(\xi -\zeta +1\right)}{2\,{\left(\zeta -1\right)}^2}\ -\frac{\zeta \,\left(2\,\zeta -2\right)\,\left(\eta -\zeta +1\right)\,\left(\xi -\zeta +1\right)}{2\,{\left(\zeta -1\right)}^2}\ \frac{\zeta \,\left(2\,\zeta -2\right)\,\left(\xi +\zeta -1\right)\,\left(\eta -\zeta +1\right)}{2\,{\left(\zeta -1\right)}^2}\ \frac{\left(2\,\zeta -1\right)\,\left(2\,\zeta -2\right)\,\left(\eta +\zeta -1\right)\,\left(\xi +\zeta -1\right)\,\left(\eta -\zeta +1\right)\,\left(\xi -\zeta +1\right)}{2\,{\left(\zeta -1\right)}^4} \end{array}\right) $$
Here we slice N3 from Pyramid5.
At $\xi = 0$: 
At $\xi = 0.01$: 
At $\xi = 0.1$: 
At $\xi = 0.5$: 
$\xi=0$ leads to a purely linear function in slice,while $\xi \neq 0$ leads to more complex distribution. Note that although singularity exists, none of them exist inside the pyramid.
The shape functions and first derivatives have singularities and discontinuities along $\zeta =0$, but for the cone inside the Pyramid, the functions seem to be regular. ?
