Q1.. Let f(x) =[a+bsinx], where x belongs to (0,Pi), a belongs to (I),b is a prime number and [.] denotes greatest integer function. Find the number of points at which f(x) is not differenciable?

Q1.. Let f(x) =[a+bsinx], where x belongs to (0,{\pi}), a belongs to (I),b is a prime number and [.] denotes greatest integer function. Find the number of points at which f(x) is not differentiable?

What does 'I' represent? Integers?

Q1.. Let , where x belongs to , a belongs to (I),b is a prime number and [.] denotes greatest integer function. Find the number of points at which f(x) is not differentiable?

I=integers

I don't know if you still need an answer or not, but I posed this
question on another math forum(mathgoodies.com) and got the following
response, which I think is very good.

I knew it would be non-differentiable at the 'jumping' point, but this is a more detailed answer.

"The derivative of a floor function at any point should be 0 if it is
differentiable at the point.

The derivative is undefined at the point where the floor function jumps.

When a+bsin(x) = I, where I is an integer, the function is not
differentiable.

Since a is an integer, b is prime, and sin(x) is between [0,1], when

sin(x)=1/b, 2/b, ...., b/b, we get a non-differentiable point. Therefore,

we have only 2b points: \left(x=sin^{-1}(\frac{1}{b}), \;\ x=sin^{-1}(\frac{2}{b})

\;\ ,......., \;\ sin^{-1}(\frac{b}{b}) and

{\pi}-sin^{-1}(\frac{1}{b}), \;\ {\pi}-sin^{-1}(\frac{2}{b}) \;\ ,......., \;\ {\pi}-sin^{-

1}(\frac{b}{b})\right) where the floor function is

not differentiable".

In other words, suppose 3 was the prime, then there would be 2b=2(3)=6
points where not differentiable: sin^{-1}(\frac{1}{3}), \;\ sin^{-1}(\frac{2}{3}), \;\ sin^{-1}(\frac{3}{3}), \;\ {\pi}-sin^{-1}(\frac{1}{3}), \;\ {\pi}-sin^{-1}(\frac{2}{3}), \;\ {\pi}-sin^{-1}(\frac{3}{3}).

Does this help? I thought this was an interesting problem. I learned something myself. I had never given any real thought to the differentiation of floor functions. I am glad you posed this problem.