(P)
The definition of function f
1 is,

The continuity of function f
1 is checked as follows.

It is clear that the function f
1 is continuous at x = 0 and NOT differentiable at x = 0 .
Hence, for option (P), (2) is correct.
(Q)
The definition of function f
2 is,

The continuity of function f
2 is checked as follows.


It is clear from the above expression that the function f
2 is NOT continuous at x = 0 .
Hence, for option (Q), (1) is correct.
(R)
The definition of function f
3 is,

The continuity of function f
2 is checked as follows.

From above expression, it is clear that, as x → 0 then, (x + 2) → 2 and thus, log
e (x + 2) → log
e2 and this is less than 1.
The range of function f
3 can be expressed as,

The definition of function f
3 can be written as,

The conclusion from the definition is,

Hence, for option (R), (4) is correct.
(S)
The definition of function f
4 is,

The continuity of function f
1 is checked as follows.

The definition of function f
4 becomes.

It is clear that the function f
4 is differentiable at x = 0 and its derivative is NOT continuous at x = 0 .
Hence, option (D) is correct.