Existing object counting methods relying on Density Map Estimation(DME) struggle with large variations in object size or input image resolution due to different imaging conditions and perspective effects. Especially, discrete grid representations of density maps result in information loss with blurred or vanished details for low-resolution inputs. To overcome these limitations, we design new Scale-Invariant Implicit Neural Representations (SI-INR) for counting to map arbitrary-scale input signals into a continuous function space, where function values over continuous spatial coordinates indicate probabilities observing objects of interest. Extensive experiments on diverse benchmark datasets have validated that SI-INR achieves robust counting performances with respect to changing input sizes, leading to better or comparable object counting accuracy compared to state-of-the-art methods.