So continuity is a much-needed required step to solve for differentiability, so I should solve for continuity first. This is what I think, differentiability and continuity are related, as we knowĪ differentiable function is always continous but not every continous function is differentiable. So, who did this right, my teacher who used the $x>1$ case in the $RHD$, or me who used the actual $x \ leq 1$ case. Answers will be posted in room 315 and 312 all week and will be. So, what my teacher asks me to do is this,Ĭalculate the $LHD$ and $RHD$ (Left hand derivative and Right hand derivative) Once we have reviewed the topic you may begin practicing the questions in your review packet. Find the values of a and b, if the function f is defined as, Our proven video lessons ease you through problems quickly, and you get tonnes of friendly practice on questions that trip students up on tests and finals. Show that, given ǫ > 0, there is a continuous g : → such that g has only finitely many fixed points and | f(x) − g(x) | < ǫ for all x ∈. For each function, determine the interval(s) of continuity. Justify for each point by: (i) saying which condition fails in the de nition of continuity, and (ii) by mentioning which type of discontinuity it is. For each graph, determine where the function is discontinuous. That is, if Y is homeomorphic to X and X has the fpp, then Y also has the fpp.Ĩ. 201-103-RE - Calculus 1 WORKSHEET: CONTINUITY 1. Prove that this prop- erty is preserved by homeomorphism. A compact topological space X has the fixed point property or fpp if every continuous self-map of X has a fixed point. Show that the Schauder Fixed Point Theorem becomes false if either of the compactness or convexity conditions does not hold.ħ. Show that the closed unit ball in the Banach space C(I,Rn) is not compact.Ħ. Prove that every linear map from R n to Rĥ. For those who know some functional analysis: is the same conclusion true for one-to-one continous onto linear maps without the assumption that there is such a k?Ĥ. Prove that there is a unique continuous linear map S : Y → X such that S(Tx) = x for all x.
2.4.5 Provide an example of the intermediate value theorem.
2.4.4 State the theorem for limits of composite functions. 2.4.2 Describe three kinds of discontinuities. Suppose T : X → Y is a one-to-one continuous onto linear map from the Banach space X to the Banach space Y and there is a constant k > 0 such that | Tx | ≥ k for all | x | = 1. Learning Objectives 2.4.1 Explain the three conditions for continuity at a point. Prove that there are constants C1 > 0, C2 > 0 such that for every x ∈ Rģ. Suppose T : X → Y is a linear map of a Banach space X into Banach space Y.
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