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Task-oriented communications, mostly using learning-based joint source-channel coding (JSCC), aim to design a communication-efficient edge inference system by transmitting task-relevant information to the receiver. However, only transmitting task-relevant information without introducing any redundancy may cause robustness issues in learning due to the channel variations, and the JSCC which directly maps the source data into continuous channel input symbols poses compatibility issues on existing digital communication systems. In this paper, we address these two issues by first investigating the inherent tradeoff between the informativeness of the encoded representations and the robustness to information distortion in the received representations, and then propose a task-oriented communication scheme with digital modulation, named discrete task-oriented JSCC (DT-JSCC), where the transmitter encodes the features into a discrete representation and transmits it to the receiver with the digital modulation scheme. In the DT-JSCC scheme, we develop a robust encoding framework, named robust information bottleneck (RIB), to improve the communication robustness to the channel variations, and derive a tractable variational upper bound of the RIB objective function using the variational approximation to overcome the computational intractability of mutual information. The experimental results demonstrate that the proposed DT-JSCC achieves better inference performance than the baseline methods with low communication latency, and exhibits robustness to channel variations due to the applied RIB framework.