Show that the sequence of functions [f_n], defined on R for each n in N as the top half of the hyperbolae
y^2 - x^2 = 1/(n^2)
converges uniformly on R to f(x) = abs(x). What do you note about the differentiability of the function at x=0?
Show that the sequence of functions [f_n], defined on R for each n in N as the top half of the hyperbolae
y^2 - x^2 = 1/(n^2)
converges uniformly on R to f(x) = abs(x). What do you note about the differentiability of the function at x=0?
See as far the convergence is concerned i havent worked out it yet. But as for the differentiablity of $\displaystyle f(x)=|x|$ is concerned you should not that the function is not differentiable at the point $\displaystyle x=0$ for the left hand derivative at $\displaystyle x=0$ is -1 whereas the R.H.D at $\displaystyle x=0$ is 1.