Existence of Tangent
Let y=f(x) be any function and
dy/dx exists at x=a
Then at the corresponding point [a , f(a)] of the function tangent to the curve exists and if the tangent
makes an angle theta with the positive direction of x-axis we have
tan(theta)=dy/dx at x=a .
tan(theta) = gradient of the tangent to the curve
y=f(x) at x=a
therefore ; if f(x) is differentiable at x=a then tangent to the curve at
x=a must exist and it must be unique .
In any graph of /x/ tangent is not unique at x=0 .
i.e. ; tangent at point x=0 when x-->0 from left is not same as the tangent
x=0 when x-->0 from right .
when x-->0 from left , gradient of the tangent is
tan135degree=-1 and when x-->0 from right , gradient of the tangent is
Hence tangent is not unique at x=0 and consequently /x/ is not differentiable at x=0 .