Convolutional neural networks have been successful in solving many socially important and economically significant problems. This ability to learn complex high-dimensional functions hierarchically can be attributed to the use of nonlinear activation functions. A key discovery that made training deep networks feasible was the adoption of the Rectified Linear Unit (ReLU) activation function to alleviate the vanishing gradient problem caused by using saturating activation functions. Since then, many improved variants of the ReLU activation have been proposed. However, a majority of activation functions used today are non-oscillatory and monotonically increasing due to their biological plausibility. This paper demonstrates that oscillatory activation functions can improve gradient flow and reduce network size. Two theorems on limits of non-oscillatory activation functions are presented. A new oscillatory activation function called Growing Cosine Unit(GCU) defined as $C(z) = z\cos z$ that outperforms Sigmoids, Swish, Mish and ReLU on a variety of architectures and benchmarks is presented. The GCU activation has multiple zeros enabling single GCU neurons to have multiple hyperplanes in the decision boundary. This allows single GCU neurons to learn the XOR function without feature engineering. Experimental results indicate that replacing the activation function in the convolution layers with the GCU activation function significantly improves performance on CIFAR-10, CIFAR-100 and Imagenette.