Show that the function `f(x) = {:{(-x^2,xle0),(x^2,x>0):}`.
is differentiable at x = 0?

Show that the function f(x) = {{:(-x^2, x le 0),(x^2 ,x gt 0):} is continuous at x = 0 . Also show that f is differentiable at x = 0 .

Show that the function f(x) = {:{(|x|, x le2),([x],x>2):} is continouous on [0,2]

If f(x) = {:{(xsin(1/x),xne0),(0,x=0):} Show that 'f' is not differentiable at x =0

Show that the function: f(x) = {:{(2+x,ifxge0),(2-x,ifx<0):} is continuous but not derivable at x=0

Discuss the continuity of the function f(x) = {:{(x^2, x le 0),(1-x,x>0):} at x = 0

Examine the continuity of the function f(x) = {:{(3x-2,xle0),(x+1,x>0):} at x = 0

Show that the function f(x) = {:{(x^n sin(1/x), x ne 0),(0, x = 0):} is continuous but not differentiable at x= 0.

Examine for continuity and differntiability, each of the following functions: f(x) = {:{(1+x,xle0),(1-x, x >0):} at x =0

Show that the function f(x) = {{:(x^2 + 3 , "if" x != 0),(1,"if" x = 0):} is not continous at x = 0 .

Prove that the function f(x) = {:{((sinx)/x , x < 0),(x^2 +1, x ge0):} is continuous.