Mizar Commentary on Walter Rudin's Principles of Mathematical Analysis/Continuity

Limits of Functions
4.1 Definition For real numbers given by LIMFUNC3:28. Else no reference.

4.2 Theorem For real numbers lim(f,p) is defined in LIMFUNC3:def 4. Else no reference.

Corollary Given by the uniqueness property of lim(f,p).

4.3 Definition $$f+g$$, $$f-g$$, $$fg$$ and $$f/g$$ are given by VALUED_1:def 1, VALUED_1:def 9, VALUED_1:def 4 and [<tt>VALUED_1:def 10</tt> or <tt>RFUNCT_1:def 1</tt> (the former maps inverse of 0 to 0, the latter removes these from the domain)] respectively. <tt>f is constant</tt> is defined in <tt>FUNCT_1:def 10</tt>, $$f=c$$ is <tt> E --> c </tt> (<tt>FUNCOP_1:def 2</tt>), with clusters in <tt>VALUED_0</tt>. $$f\geq g$$ is defined in <tt>COUSIN2:def 3</tt>. Addition of functions yielding complex vectors is given by <tt>INTEGR15:def 9</tt> and <tt>VALUED_2:def 45</tt>, component-wise multiplication by <tt>VALUED_2:def 47</tt>. The clustering for the vectors is given in <tt>TOPREALC</tt>. Scalar multiplication is given by <tt>VALUED_1:def 5</tt>, with correct clustering in the same file.

4.4 Theorem For reals: (a) is <tt>LIMFUNC3:33</tt>. (b) is <tt>LIMFUNC3:38</tt>. (c) is <tt>LIMFUNC3:39</tt>. Else no references.

Remark No reference.

Continuous Functions
4.5 Definition Continuity of a function between topological spaces in a point is defined in <tt>TMAP_1:def 2</tt>, on the whole domain in <tt>TMAP_1:44</tt> (see also <tt>PRE_TOPC:def 6</tt>). No reference for metric spaces. For real functions given by <tt>FCONT_1:3</tt> in a point and by <tt>FCONT_1:def 2</tt> for whole domain. For complex functions given by <tt>CFCONT_1:32</tt> in a point and by <tt>FCONT_1:def 2</tt> for a domain. For real vectors by <tt>NFCONT_1:7</tt> (see also <tt>PDIFF_7:def 6</tt>) in a point and by <tt>NFCONT_1:def 7</tt> for a domain.

4.6 Theorem No direct reference, in spirit given by the referred theorems under 4.5 in combination with the definitions using sequences (see <tt>FCONT_1:def 1</tt>, <tt>CFCONT_1:def 1</tt> and <tt>NFCONT_1:def 5</tt>).

4.7 Theorem For topological spaces given by <tt>TMAP_1:47</tt> (pointwise) and <tt>TOPS_2:46</tt> (whole domain). For real functions given by <tt>FCONT_1:12</tt> (pointwise) and clustered there for whole domain. Else no reference.

4.8 Theorem For topological spaces given by <tt>TOPS_2:43</tt>. For real functions and real vectors only given in neighborhood variant (<tt>FCONT_1:5</tt> and <tt>NFCONT_1:9</tt> (the latter only pointwise)). No reference else.

Corollary is given by original definition <tt>PRE_TOPC:def 6</tt> for topological spaces, else no reference.

4.9 Theorem For real functions given by <tt>FCONT_1:7</tt> and <tt>FCONT_1:11</tt> (pointwise) and clustered there, except for $$f/g$$ (<tt>FCONT_1:24</tt>) (whole domain). For complex functions given by <tt>CFCONT_1:33</tt> and <tt>CFCONT_1:37</tt> (pointwise) and by <tt>CFCONT_1:43</tt> and <tt>CFCONT_1:49</tt> on domain.

4.10 Theorem (a) Only special case where the metric space is $$R$$ in <tt>NFCONT_4:43</tt>. (b) Only addition in <tt>NFCONT_1:15</tt>.

4.11 Examples The projections are defined in <tt>PDIFF_1:def 1</tt>. No reference for the continuity of general polynoms or rational functions. $$|x|$$ is continuous by <tt>NFCONT_1:17</tt> (pointwise) and <tt>NFCONT_1:28</tt> (domain).

4.12 Remark Nothing to formalize.

Continuity and Compactness
4.13 Definition For real functions given by <tt>RFUNCT_1:72</tt>, for complex functions given by <tt>SEQ_2:def 5</tt>. For functions into $$R^k$$ implicitly given by <tt>INTEGR19:14</tt> (the left <tt>bounded</tt> is from <tt>INTEGR15:def 12</tt>) which basically uses the maximum norm (compare <tt>NFCONT_4:def 2</tt>). No direct reference for the Euclidean norm.

4.14 Theorem For topological spaces more generally given by <tt>COMPTS_1:15</tt> or <tt>WEIERSTR:8</tt>. For real functions given by <tt>FCONT_1:29</tt>, for complex functions by <tt>CFCONT_1:52</tt>. For real vectors given by <tt>NFCONT_1:32</tt>.

4.15 Theorem In <tt>PSCOMP_1</tt> clustered via use of pseudocompactness of topological spaces (<tt>PSCOMP_1:def 4</tt>). No reference for metric spaces. For real domain directly given by <tt>INTEGRA5:10</tt>.

4.16 Theorem In <tt>PSCOMP_1</tt> clustered via use of pseudocompactness of topological spaces (<tt>PSCOMP_1:def 4</tt>), again. No reference for metric spaces as well. For real domain directly given by <tt>FCONT_1:30</tt>.

4.17 Theorem No reference. Next best variant for real functions would be <tt>FCONT_3:22</tt>.

4.18 Definition For metric spaces given by <tt>UNIFORM1:def 1</tt>. For real functions given by <tt>FCONT_2:def 1</tt>. For vectors given by <tt>NFCONT_2:def 1</tt>, see also <tt>INTEGR20:def 1</tt> and the definitions in <tt>NCFCONT2</tt>. That uniform continuous functions are continuous is given by <tt>UNIFORM1:5</tt> for metric spaces (indirectly), by <tt>FCONT_2:9</tt> for real functions and by <tt>NFCONT_2:7</tt> for vectors.

4.19 Theorem For metric spaces indirectly given by <tt>UNIFORM1:7</tt>. For real functions given by <tt>FCONT_2:11</tt>, for vectors given by <tt>NFCONT_2:10</tt>.

4.20 Theorem (a) No reference. (b) No reference. (c) No reference.

4.21 Example $$f(2\pi t)$$ as a function defined on the real line is given by <tt>CircleMap</tt> in <tt>TOPREALB:def 11</tt>. The continuity of that function as well as its surjectivness and injectiveness (the latter only if restricted to a half-open interval of length 1) is clustered there as well. No reference for the inverse explicitly not being continuous.

Continuity and Connectedness
4.22 Theorem Given more generally for topological spaces by <tt>TOPS_2:61</tt>. No reference specifically for real functions.

4.23 Theorem <tt>TOPREAL5:8</tt>, generalized version is <tt>TOPREAL5:5</tt>. No reference for version with simple real functions. #TODO Find simple version of intermediate value theorem.

4.24 Example No reference.

Discontinuities
4.25 Definition $$f(x\pm)$$ is <tt>lim_right/left(f,x)</tt> (<tt>LIMFUNC2:def 8/7</tt>). The "It is clear" is <tt>LIMFUNC3:29/30</tt>.

4.26 Definition No direct reference, although the properties are indirectly given: $$f$$ having a discontinuity of the first kind at $$x$$ means <tt>not f is_convergent_in x & f is_left_convergent_in x & f is_right_convergent_in x</tt>; $$f$$ having a discontinuity of the second kind at $$x$$ means <tt>not f is_left_convergent_in x or not f is_right_convergent_in x</tt>.

4.27 Examples (a) $$f$$ is given by <tt>(REAL --> 1) - chi(RAT,REAL)</tt> (see <tt>FUNCOP_1:def 2</tt> and <tt>FUNCT_3:def 3</tt>). No reference for the discontinuity property. (b) $$f$$ is given by <tt>(id REAL)*((REAL --> 1) - chi(RAT,REAL))</tt> (see additionally RELAT_1:def 10 and the functional properties in <tt>FUNCT_1</tt>). No reference for the discontinuity property. (c) No reference. (d) $$f$$ is given by <tt>sin((id REAL)^) +* (0 .--> 0)</tt> (see <tt>SIN_COS:def 16</tt>, <tt>RFUNCT_1:def 2</tt>, <tt>FUNCT_4:def 1</tt> and <tt>FUNCOP_1:def 9</tt>). No reference for the discontinuity property.

Monotonic Functions
4.28 Definition Monotonically increasing/decreasing is <tt>non-decreasing/increasing</tt> (<tt>VALUED_0:def 15/16</tt>). The <tt>monotone</tt> property is in <tt>RFUNCT_2:def 5</tt>.

4.29 Theorem No reference, although <tt>FCONT_3</tt> is full of related result.

Corollary No reference.

4.30 Theorem No reference. #TODO Find proof that number of discontinuities of a monotone function are at most countable.

4.31 Remark No reference.

Infinite Limits and Limits at Infinity
4.32 Definition No reference, although open intervals were defined in Mizar with infinity in mind.

4.33 Definition Limits at infinity are explicitly formalized as <tt>lim_in+infty f</tt> and <tt>lim_in-infty f</tt> (<tt>LIMFUNC1:def 12/13</tt>).

4.34 Theorem <tt>uniqueness</tt> is given again by property of definitions <tt>LIMFUNC1:def 12/13</tt>. $$f+g$$, $$fg$$ and $$f/g$$ are given by <tt>LIMFUNC1:82/91</tt>, <tt>LIMFUNC1:87/96</tt> and <tt>LIMFUNC1:88/97</tt> respectively.

Exercises
1. No reference. 2. No reference, including no counterexample for the backwards direction. #TODO Show that $$f(\overline{E})\subset\overline{f(E)}$$. 3. $$Z(f)$$ is usually referred to as <tt>f"{0}</tt> (<tt>FUNCT_1:def 7</tt>). No direct reference, but pretty obvious even by Mizar standard. Because of <tt>PRE_TOPC:def 6</tt> we only need to show that <tt>{0}</tt> is closed in <tt>R^1</tt> and we get that from clustering <tt>R^1 -> T_2 -> T_1</tt> (<tt>TOPREAL6</tt> and <tt>PRE_TOPC</tt>) and <tt>URYSOHN1:19</tt>. 4. No reference. #TODO Proof that images of dense sets under continuous functions are dense and that continuous functions are determined by their dense subsets. 5. No reference. Basically the simple case of 4 for real functions. 6. No reference. Since functions are relations in Mizar, $$f$$ consists of elements of the form <tt>[x,f.x]</tt>. However, for a proper embedding into <tt>TOP-REAL 2</tt> one would need the set of all <tt>&lt;*x,f.x*&gt;</tt>. 7. $$g$$ is referred to as <tt>f|E</tt> (<tt>RELAT_1:def 11</tt>). No reference for the exercise. 8. No reference. 9. No reference. 10. No reference. 11. No reference. 12. No reference. #TODO Compositions of uniformly continuous functions are uniformly continuous. 13. No reference. 14. No reference. #TODO Existence of fixpoint for continuous $$f:[0,1]\rightarrow[0,1]$$. 15. <tt>open</tt> is defined in <tt>T_0TOPSP:def 2</tt>. No reference for exercise. 16. $$[x]$$ is <tt>[\x/]</tt> (<tt>INT_1:def 6</tt>), $$(x)$$ is <tt>frac x</tt> (<tt>INT_1:def 8</tt>). No reference for exercise. 17. No reference. 18. No reference. 19. No reference. 20. $$\rho_E(x)$$ is given by <tt>dist(x,E)</tt> (<tt>SEQ_4:def 17</tt> (complex vectors), <tt>JORDAN1K:def 2</tt> (real vectors)). No references for exercises. 20.(a) No reference, but see <tt>SEQ_4:116</tt> and <tt>JORDAN1K:45</tt>. 20.(b) No reference. 21. No reference. 22. No reference. 23. No reference. For convexity see <tt>CONVFUN1:4</tt>. 24. No reference. 25. No reference. 26. No reference.