  Repeating fast radio bursts can exhibit a wide range of burst repetition
rates, from none to hundreds of bursts per hour. Here, we report the detection
and characteristics of 57 bursts from the recently discovered FRB 20240114A,
observed with GMRT in the frequency ranges 300-500 MHz and 550-750 MHz.
Majority of the bursts show narrow emission-bandwidth with $\Delta\nu/\nu \sim$
around 10 %. All of the bursts we detect are faint ($<$10 Jy ms), and thus
probe the lower end of the energy distribution. We determine the rate function
for FRB 20240114A at 400 MHz, and downward drift rates at 400 and 650 MHz, and
discuss our measurements in the context of the repeating FRB population. We
observe sudden variations in the burst activity of FRB 20240114A over time. Our
data as well as the other publicly available information on other observations
of FRB 20240114A so far, there is an indication that FRB 20240114A potentially
exhibit a chromaticity in its burst activity. While the burst properties of FRB
20240114A are similar to ther repeating FRBs, the frequency-dependent activity,
if established, could provide crucial clues to the origin of repeating FRBs. We
also place the most stringent 5$\sigma$ upper limits of 600 $\mu$Jy and 89
$\mu$Jy on any persistent radio source (PRS) associated with FRB 20240114A at
400 MHz and 650 MHz, respectively, and compare these with the luminosity of the
known PRSs associated with FRB121102A and FRB190520B.
