A vast and growing number of IoT applications connect physical devices, such as scientific instruments, technical equipment, machines, and cameras, across heterogenous infrastructure from the edge to the cloud to provide responsive, intelligent services while complying with privacy and security requirements. However, the integration of heterogeneous IoT, edge, and cloud technologies and the design of end-to-end applications that seamlessly work across multiple layers and types of infrastructures is challenging. A significant issue is resource management and the need to ensure that the right type and scale of resources is allocated on every layer to fulfill the application's processing needs. As edge and cloud layers are increasingly tightly integrated, imbalanced resource allocations and sub-optimally placed tasks can quickly deteriorate the overall system performance. This paper proposes an emulation approach for the investigation of task placements across the edge-to-cloud continuum. We demonstrate that emulation can address the complexity and many degrees-of-freedom of the problem, allowing us to investigate essential deployment patterns and trade-offs. We evaluate our approach using a machine learning-based workload, demonstrating the validity by comparing emulation and real-world experiments. Further, we show that the right task placement strategy has a significant impact on performance -- in our experiments, between 5% and 65% depending on the scenario.