A differentiable function at a point in Banach spaces is defined in terms of the existence of a linear approximation that closely resembles the function near that point. The discussion highlights a lack of clarity in existing resources, including Wikipedia, which may not align with specific definitions used in certain contexts. Users express difficulty in finding authoritative texts that provide a clear definition. One participant mentions finding a definition from an obscure source, indicating the variability in available literature. The conversation underscores the need for precise definitions in mathematical contexts, particularly in advanced topics like Banach spaces.