Sample Questions for Exam 1 (Limits – Sections 2.1 to 2.5) 1. Sketch
... Investigate the limit of f as x approaches 1. Does the limit exist? If so, justify your answer. ...
... Investigate the limit of f as x approaches 1. Does the limit exist? If so, justify your answer. ...
Section 3.1
... derivative either is zero or does not exist. The x-values at these special points are called critical numbers. Figure 3.4 illustrates the two types of critical numbers. ...
... derivative either is zero or does not exist. The x-values at these special points are called critical numbers. Figure 3.4 illustrates the two types of critical numbers. ...
Definition: lim f(x) = L means: (1) f is defined on an open interval
... In order to prove that every ǫ > 0 has property P , as in the chart above, the proof must start with “Let ǫ be an arbitrary positive real number.” This is somewhat verbose, and we will abbreviate it (standardly) to “Let ǫ > 0.” We did not change the meaning with this abbreviation. After this start, ...
... In order to prove that every ǫ > 0 has property P , as in the chart above, the proof must start with “Let ǫ be an arbitrary positive real number.” This is somewhat verbose, and we will abbreviate it (standardly) to “Let ǫ > 0.” We did not change the meaning with this abbreviation. After this start, ...
Lecture 4: examples of topological spaces, coarser and finer
... The set BR of open intervals in R satisfies this condition: the intersection of open intervals is always an open interval (possibly empty). So it’s a basis. Example 6. An open disc in R2 is the collection of points less than distance r away from a fixed point (x0 , y0 ): D = {(x, y) ∈ R2 | (x − x0 ) ...
... The set BR of open intervals in R satisfies this condition: the intersection of open intervals is always an open interval (possibly empty). So it’s a basis. Example 6. An open disc in R2 is the collection of points less than distance r away from a fixed point (x0 , y0 ): D = {(x, y) ∈ R2 | (x − x0 ) ...
Between strong continuity and almost continuity
... (d) extremally disconnected topology if the closure of every open set in X is open in X. It turns out that the notions of almost partition topology and extremally disconnected topology are identical notions. Moreover, partition topology ⇒ δ-partition topology ⇒ almost partition topology (≡ extremall ...
... (d) extremally disconnected topology if the closure of every open set in X is open in X. It turns out that the notions of almost partition topology and extremally disconnected topology are identical notions. Moreover, partition topology ⇒ δ-partition topology ⇒ almost partition topology (≡ extremall ...
Stability of convex sets and applications
... complete version was proved by O’Brien [5]. In what follows this result is referred to as the Vesterstrøm–O’Brien theorem. It does not hold for general non-compact convex sets, but it does hold for convex µ-compact sets ([10], Theorem 1), which are defined as follows. Definition 2. A convex set A is ...
... complete version was proved by O’Brien [5]. In what follows this result is referred to as the Vesterstrøm–O’Brien theorem. It does not hold for general non-compact convex sets, but it does hold for convex µ-compact sets ([10], Theorem 1), which are defined as follows. Definition 2. A convex set A is ...
Metric Spaces in Synthetic Topology
... Up to topological equivalence, a set has at most one complete separable metric (which then induces the intrinsic topology). ...
... Up to topological equivalence, a set has at most one complete separable metric (which then induces the intrinsic topology). ...
