  We consider the problem of detecting gradual changes in the sequence of mean
functions from a not necessarily stationary functional time series. Our
approach is based on the maximum deviation (calculated over a given time
interval) between a benchmark function and the mean functions at different time
points. We speak of a gradual change of size $\Delta $, if this quantity
exceeds a given threshold $\Delta>0$. For example, the benchmark function could
represent an average of yearly temperature curves from the pre-industrial time,
and we are interested in the question if the yearly temperature curves
afterwards deviate from the pre-industrial average by more than $\Delta =1.5$
degrees Celsius, where the deviations are measured with respect to the
sup-norm. Using Gaussian approximations for high-dimensional data we develop a
test for hypotheses of this type and estimators for the time where a deviation
of size larger than $\Delta$ appears for the first time. We prove the validity
of our approach and illustrate the new methods by a simulation study and a data
example, where we analyze yearly temperature curves at different stations in
Australia.
