A Brownian non-differentiable monster is a continuous function of space-time that is also nowhere differentiable. In many cases, functional continuity never implies differentiability. Continuous functions with no derivatives were studied in the middle of the 19th century. The classical example that attracted the most attention was the one affirmed by Karl Weierstrass (1815-97) in a lecture to the Berlin Academy on July 18, 1872. He communicated his example in a letter to another German mathematician Paul Du Bois-Reymond (1831-89) who invented the term integral equations, and it was published by the latter. The function is given by ��(��)=∑��ⁿ������(��ⁿ����) as the power index varies from zero to infinity where �� is an odd integer and �� is a positive constant less than unity such that ����>1+(3��/2). This infinite series is uniformly convergent and so defines a continuous function. Weierstrass’s example prompted the creation of many more functions that are continuous in an interval or everywhere but fail to be differentiable either on a dense set of points or at any point. The great historical significance of the discovery that continuity does not imply differentiability and that functions can have all sorts of abnormal behavior made mathematicians all the more wary and fearful of trusting intuition or geometrical thinking concerning the existence of mathematical untamed monsters. Reference: Morris Kline, Mathematical Thought from Ancient to Modern Times, p956, Oxford University Press, 1972.

A modern day monster curve takes the form of a continuous function describing the Brownian motion of space-time charges of squares of zero-point energies of the quantum vacuum fluctuations. Fortunately, slaying or taming this monster curve is simply to set 2��ⁿ�� to an odd integer such that the cosine argument is always an odd multiple of ��/2 or equivalently odd multiples of 90°, implying orthogonality of the corresponding vector field and consequently the corresponding direction cosines are always zero and hence the function itself is also zero regardless of the values of the �� or ��ⁿ factors. Clearly, Weierstrass’s monster curve is a partial sum of an expanded quadratic form of infinitely many variables wherein only the specified direction cosines are shown. Completing the partial sums requires adding Hilbert spaces. However, these square integrable partial sums are uniformly divergent unless alternating positive and negative squares of energy exist.