Multivariate piecewise fxn continuity and partial derivative

[vchurchill]

1. Problem

Define a function:

for t>=0, f(x,t) = { x for 0 <= x <= sqrt(t), -x + 2sqrt(t) for sqrt(t) <= x <= 2sqrt(t), 0 elsewhere}

for t<0 f(x,t) = - f(x,|t|)

Show that f is continuous in R^2. Show that f_t (x, 0) = 0 for all x.

Then define g(t) = integral[f(x,t)dx] from -1 to 1. Show g(t) = t for |t|<1/4, and hence g'(0) != integral[f(x,0)dx] from -1 to 1.

2. Solution attempt

For the first part, I used an epsilon-delta proof:

For all epsilon > 0 there exists 0<delta=epsilon/3 s.t. |x-x0|<eps/3 implies |f(x)-f(x0)|<eps. Checking for both important pieces of f, this works.

Then I computed f_t(x,t) = { 1/sqrt(t) for sqrt(t) <= x <= 2sqrt(t), and 0 elsewhere}. I do not understand how f_t(x,0) then is equal to 0 for all x since it would seem that for x = 0 we have f_t(0,0) = 1/sqrt(0) = ??

Then the last part... the piecewise-ness of this fxn makes it extremely difficult to conceptualize and get a general g(t), so I'm guessing I should double integrate over -1/4 to 1/4 for t and -1 to 1 for x. Does this make sense?

Thanks for your help.

 

[Svein]