Consider the following statements :
S 1 : Let f(x) = 
, where [ . ] stands for the greatest integer function. Then f(x) is discontinuous at x = n + π , n ∈ Ι
S 2 : The function f(x) = p[x + 1] + q [x – 1], (where [.] denotes the greatest integer function) is continuous at x = 1 if p + q = 0
S 3 : Let f(x) = |[x] x| for – 1 ≤ x ≤ 2, where [.] is greatest integer function, then f is not differentiable at x = 2.
S 4 : If f(x) takes only rational values for all real x and is continuous, then f ′ (10) = 10.
State, in order, whether S 1 , S 2 , S 3 , S 4 are true or false
Text Solution
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S 1 : f(x) = 
[x – π ] is an integer for x ∈ R
∴ f(x) = 0 
x ∈ R.
Hence f(x) is always continuous. (False)
S 2 : f(x) = p[x + 1] + q [x – 1]
= (p + q) [x] + p – q
f(1) = 2p
f(1 + ) = 2p
f(1 – ) = p – q
But f(x) is continous at x = 1
2p = p – q p + q = 0 [True]
S 3 : f(x) = |[x] x| = 
function is not continuous at x = 2
∴ non-differentiable also (True)
S 4 : f(0) = constant
f ′ (0) = 0 
f ′ (10) = 0 [False]
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