Determine the haar wavelet function is linear or nonlinear
Linearity can play at different levels in this context:

The Haar wavelet functions are piece-wise affine (constant), not linear (but sometimes people call affine &quot;linear&quot;).
Any standard wavelet decomposition is linear, as it decomposes a signal into a sum of coefficients times a wavelet. So this is not specific of the Haar wavelet.
Wavelet approximations are often nonlinear, as they consist in choosing the $K$ best (largest) coefficients to represent the signal. And choosing the $K$ largest does not obey the linearity principle.

So I cannot answer directly, by lack of context.