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Here is a picture of a function that is continuous everywhere and whose first partials exist everywhere, but the first partials are discontinuous at the origin. The function is z=(x^4y+4x^2y^3-y^5)/(x^2+y^2)^2 (defined to be zero at the origin). The basic shape is that of a monkey saddle (like a regular saddle but with an extra slot for the tail), except that it's a bit pointier at the origin than the standard monkey saddle.
The following picture is a view straight down the x-axis. The idea is to show you that the partial with respect to x at the origin is zero (the surface goes right along the axis, straight toward your face) and the partial with respect to y at the origin is -1 (the left-to-right downward slant).
The following picture shows the surface again along with the plane representing the linearization at the origin. The blue and green lines are the lines tangent to the surface whose slopes are the partial derivatives. As determined by these lines, the linearization plane has equation L(x,y)=0(x-0)-(y-0)+0=-y. You can see that the nearest blue "fin" of the saddle comes upward out of the origin, while the plane heads down in that direction (the direction is that of the line y=x).
Here's a better view of the surface and the plane parting ways. The red line is the line y=x lifted up to the surface (where it remains a line), while the yellow line is the line y=x lifted up to the plane. Clearly they are headed in very different directions. The function is not differentiable at the origin.
