DAT301

• The common single variable function \(f(x)\) is a single output dependent on a single input.
• A two-variable function is when a single output is dependent on two inputs:

\[ z = f(x, y) \]

• In 2-dimensions the graph of \(f(x)\) is a line on the \(xy\)-plane

• The graph of \(z = f(x,y)\) is the set of points \((x,y,f(x,y))\) in 3D space.
• It is a surface sitting above or below the \(xy\) plane.

x <- seq(-3, 3, length.out = 100)
y <- seq(-3, 3, length.out = 100)
z <- outer(x, y, function(x, y) x^2+y^2)

persp(x, y, z, theta = 30, phi = 20, col="red", shade=0.5, xlab="x-axis", 
      ylab="y-axis", zlab="z-axis", 
      main="3D Function: z = f(x, y) = x^2 + y^2")

• If any vertical line (parellel to the z-axis) intersects the surface at more than one point, the the surface fails to be a function.
• For each \((x,y)\) value in the circular base; there are two possible z-values, one on the upper hemishpere (blue), and on on the lower hemishphere (pink).

• A linear function: \[ z = ax + by +c \]

is a plane. Its slope in each direction is determined by \(a\) and \(b\).

• When working with two dimensions we can approach a point from infinitely many directions, thus we say:

\[ \lim_{(x, y) \to (a, b)} f(x, y) = L \]

• Since we can approach \((a,b)\) in many ways, we often test the limit along particular paths (curves through the point).
• If the limit exists, it MUST be the same along every possible path approaching the point.
• Thus we can prove a limit doesn’t exist by finding a single path that doesn’t conform; this is proof by contradiction.

• Continuity at \(a\) requires that:

1. \(f(a,b)\) is defined
2. \(\lim_{(x, y) \to (a, b)} f(x, y)\) exists.
3. They are equal to: \[ \lim_{(x, y) \to (a, b)} f(x, y) = f(a,b) \]

• Even if a surface looks like it is connected, it may still be discontinuos at certain points. Continuity is about behavior of fucntion values, not visuals.