Partial Derivatives (Section 13.3).

Continuous function: z = 4-(x^2)-2(y^2)

g[x_,y_]:= 4-x^2-2y^2
Plot3D[g[x,y],{x,-2,2},{y,-2,2},Mesh->None,PlotStyle->{Opacity[0.8],Pink}]

Discontinuous function at (0,0): z = xy/(x^2+y^2)

h[x_,y_]:= ((x*y)/(x^2+y^2))
Plot3D[h[x,y],{x,-2,2},{y,-2,2},Mesh->None,PlotStyle->{Opacity[0.8],Green}]

Partial derivatives interpretation: slopes of curves of intersection.
Function z = 4-(x^2)-2(y^2) and its partial derivatives at the point (1,1,1).

g[x_,y_]:= 4-x^2-2y^2
(* this is how you can compute partial derivatives *)
Dgx[x_,y_]:=D[g[x,y],x] (* partial derivative of g w.r.t. x *)
Dgy[x_,y_]:=D[g[x,y],y] (* partial derivative of g w.r.t. y *)
Dgx[x,y]
Dgy[x,y]

Show[
(* plane y=1; we fix y to get partial derivative w.r.t. x *)
ContourPlot3D[y==1, {x,-2,2},{y,-2,2},{z,-5,5}, Mesh->None, ContourStyle->Opacity[0.5],AxesLabel->{x,y,z}],
(* the surface *)
Plot3D[g[x,y],{x,-2,2},{y,-2,2},Mesh->None,PlotStyle->{Opacity[0.5],Pink}],
(* curve of intersection of the surface z=g(x,y) with the plane y=1 *)
ParametricPlot3D[{t,1,g[t,1]},{t,-2,2},PlotStyle->{Green}] ,
(* tangent lint to the curve of intersection *)
ParametricPlot3D[{1+s,1,1-2s},{s,-2,2},PlotStyle->{Black}],
Graphics3D[{PointSize[0.02], Point[{1,1,g[1,1]}]}],
Graphics3D[{Text[Style["slope ((∂g)/(∂x))(1,1)",Bold]]},{1,1,1}]
]

Show[
(* plane x=1; we fix x to get partial derivative w.r.t. y *)
ContourPlot3D[x==1, {x,-2,2},{y,-2,2},{z,-5,5}, Mesh->None, ContourStyle->Opacity[0.5],AxesLabel->{x,y,z}],
(* the surface *)
Plot3D[g[x,y],{x,-2,2},{y,-2,2},Mesh->None,PlotStyle->{Opacity[0.5],Pink}],
(* curve of intersection of the surface z=g(x,y) with the plane x=1 *)
ParametricPlot3D[{1,t,g[1,t]},{t,-2,2},PlotStyle->{Green}] ,
(* tangent lint to the curve of intersection *)
ParametricPlot3D[{1,1+s,1-4s},{s,-2,2},PlotStyle->{Black}],
Graphics3D[{PointSize[0.02], Point[{1,1,g[1,1]}]}],
Graphics3D[{Text[Style["slope ((∂g)/(∂y))(1,1)",Bold]]},{1,1,1}]
]

Example of a piecewise function f(x,y) which is not continuous, but has partial derivatives at (0,0):
illustration of the necessity to have a stronger condition to connect partial derivatives and continuity!

f[x_,y_]:=Piecewise[{{1,x*y==0},{0,True}}]
Show[
Plot3D[f[x,y],{x,-5,5},{y,-5,5},Mesh -> None, AxesLabel->{"x","y","z"}],
ParametricPlot3D[{x=t,y=0,z=1},{t,-5,5}],
ParametricPlot3D[{x=0,y=t,z=1},{t,-5,5}]
]