What is the difference between differentiable and continuous Differentiability is a stronger condition than continuity If f is differentiable at x = a, then f is continuous at x = a as well But the reverse need not hold Continuity of f at x = a requires only that f(x) − f(a) converges to zero as x → a For differentiability, that difference is required to converge even after being divided by x − a In other words, f(x) − f(a) x − a must
What is a continuous stochastic process? - Mathematics Stack Exchange So, there will be a discontinuity at time k Isn't this violating the definition of continuous stochastic process or is it that I have to keep ω ω constant throught out the process ? Also, is ω ω in the definition of continuous stochastic process the outcome at any point of time or is it the string of the outcomes that occurs till time
Proving the inverse of a continuous function is also continuous 6 All metric spaces are Hausdorff Given a continuous bijection between a compact space and a Hausdorff space the map is a homeomorphism Proof: We show that f f is a closed map Let K ⊂E1 K ⊂ E 1 be closed then it is compact so f(K) f (K) is compact and compact subsets of Hausdorff spaces are closed Hence, we have that f f is a homeomorphism