Tuesday and Wednesday's lectures were about continuity (4.2).
Thursday's lecture was about partial derivatives (4.3).
Monday's lecture will be about differentiability (4.4).
We can literally see the difference between these concepts.
1. Here is a function that has x and y partial derivatives
at every point but is discontinuous at (0,0):
f(x,y)=xy/(x^2+y^2) if (x,y) not (0,0) else 0.
http://www.wolframalpha.com/input/?i=xy%2F%28x^2%2By^2%29
This function appears in HW20. As (x,y) approaches
(0,0) along the x or y axis, f(x,y) stays zero.
If we approach (0,0) along a slanted line:
http://dkmj.org/2415s16/ipython/continuity.html
(Scroll down to the interactive images.)
2. Here is a function that is continuous at every point
but has neither x nor y partial derivative at (0,0):
f(x,y)=(x^2+y^2)^(1/2).
You should recognize this as a cone.
The vertex has no tangent line.
http://www.wolframalpha.com/input/?i=sqrt%28x^2%2By^2%29
3. Here is a function that is continuous at every point
and has x and y partial derivatives at every point,
but is not differentiable (see section 4.4) at (0,0):
f(x,y)=x^3/(x^2+y^2) if (x,y) not (0,0) else 0.
http://www.wolframalpha.com/input/?i=x^3%2F%28x^2%2By^2%29
z=f(x,y) has tangent lines at (0,0,0) in the x and y directions
and in infinitely many other directions,
but these tangent lines do not lie in the same plane,
so there is no tangent plane at (0,0,0):
http://dkmj.org/2415s16/ipython/differentiability.html
(Scroll down to the interactive images.)