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The continuous wavelet transform (CWT) is a frequently used tool to study periodicity in climate and other time series. Periodicity plays a significant role in climate reconstruction and prediction. In numerous studies, the use of CWT revealed dominant periodicity (DP) in climatic time series. Several studies suggested that these "natural oscillations" would even reverse global warming. It is shown here that the results of wavelet analysis for detecting DPs can be misinterpreted in the presence of local singularities that are manifested in lower frequencies. This may lead to false DP detection. CWT analysis of synthetic and real-data climatic time series, with local singularities, indicates a low-frequency DP even if there is no true periodicity in the time series. Therefore, it is argued that this is an inherent general property of CWT. Hence, applying CWT to climatic time series should be reevaluated, and more careful analysis of the entire wavelet power spectrum is required, with a focus on high frequencies as well. A conelike shape in the wavelet power spectrum most likely indicates the presence of a local singularity in the time series rather than a DP, even if the local singularity has an observational or a physical basis. It is shown that analyzing the derivatives of the time series may be helpful in interpreting the wavelet power spectrum. Nevertheless, these tests are only a partial remedy that does not completely neutralize the effects caused by the presence of local singularities.