Limits Continuity And Differentiability

Let f : R \(\to\) R be defined as

\(f(x) = \left[ {\matrix{ {[{e^x}],} & {x < 0} \cr {a{e^x} + [x - 1],} & {0 \le x < 1} \cr {b + [\sin (\pi x)],} & {1 \le x < 2} \cr {[{e^{ - x}}] - c,} & {x \ge 2} \cr } } \right.\)

where a, b, c \(\in\) R and [t] denotes greatest integer less than or equal to t. Then, which of the following statements is true?