Let f ( x ) = { | x | x if x ≠ 0 1 if x = 0 (a) Show that f is continuous at 0. (b) Investigate graphically whether f is differentiable at 0 by zooming in several limes toward the point (0, 1) on the graph of f . (c) Show that f is not differentiable at 0. How can you reconcile this fact with the appearance of the graphs in part (b)?
Let f ( x ) = { | x | x if x ≠ 0 1 if x = 0 (a) Show that f is continuous at 0. (b) Investigate graphically whether f is differentiable at 0 by zooming in several limes toward the point (0, 1) on the graph of f . (c) Show that f is not differentiable at 0. How can you reconcile this fact with the appearance of the graphs in part (b)?
Solution Summary: The author explains how to show that the function f is continuous at 0 by showing that undersetxto
Show that f is continuous but not differentiable at x=1
Assume g is continuous from the right at x_0, g(x_0) is an interior point of the domain of f and f is continuous at g(x_0). Show that f of g= f(g(x)) is continuous from the right at x_0.
What if we want to prove it is continuous from the left?
Determine if g is continuous at x=0 and if g is differentiable at x = 0 if,
let g(x) = {x2sin(1/x) if x ≠ 0}
let g(x) = 0 if x=0
Please show all steps and explain with as much detail as possible.
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