Text in red are my additions.
Mat 5:13 You are the salt of the earth. But if the salt lose its savour, wherewith shall it be salted? It is good for nothing anymore but to be cast out, and to be trodden on by men.
Here in these verse we find the influence the citizens of the Kingdom are to have on the world. (A) they are to preserve men; (B) they are to guide men; (I’ve here paraphrased and expanded a brief notation by Fr. Maas).
(A) They are to preserve men. In this section must be noted the opposition between the apostolic and the prophetic work; the meaning of “salt” as applied to the apostles; and finally, the uselessness of the salt if it loses its savor.
 The opposition between the apostles and prophets consists in the universal character of the apostolic mission as compared with the particular character of the prophetic work. Hence the apostles are the salt of the earth.
 The meaning of “salt” in this context is determined by its use in common life: it prevents putrefaction and it seasons; there seems to be no direct reference to its fertilizing property [Schanz]. In the Old Testament, too, salt had been employed for this double purpose: cf. 2 Kings 2:21; Lev. 2:13; Num. 18:19; 2 Chron13:5; Paschasius Radbertus, Maldonado, Chromatius The moral corruption of the world was therefore to be remedied by the apostolic salt of the disciples.
 As to the uselessness of the salt after it has lost its savor, many commentators regard the language of our Lord as purely hypothetical, because, according to them, salt cannot really lose its savor. But Schöttgen [Hor. hebr. i. 1.], Thomson [The Land and the Book, p. 381], and others maintain that the salt of Palestine actually does lose its savor when in contact with the ground, or exposed to rain and sun; Dr. Thomson adds that in Sidon he saw “large quantities of it literally thrown into the street, to be trodden under foot of men and beasts.” Even Pliny speaks of “sal iners” [xxxi. 39] and “sal tabescens” [xxxi. 44]. The practical impossibility of salting salt, and reducing it to its proper condition, furnishes an illustration of the dangerous condition of the disciples of Jesus that neglect their call to salt the earth, and preserve their fellow men from moral corruption [cf. Heb. 6:4 f.; Heb 10:26–29; Ezek 15:2–5; Isa 66:24; Dan 12:2; Jerome, Paschasius Radbertus, Cajetan, Jansenius Maldonado, Lamy, etc.].
 It has been supposed in the preceding remarks that Jesus addressed his apostles or his disciples directly in the text thus far explained. This must be understood in such a manner that he addressed them principally, though what he said to them applies truly, though less emphatically, to all Christians [cf. Knabenbauer Paschasius Radbertus, Chromatus]. Neither the direct address nor the comparison with salt and light forces us to admit that our Lord speaks to the disciples alone either from the beginning of v. 11 [Chrysostom, Hilary, Jerome, Rabanus Maurus, Bruno, Zach., Thomas Aquinas, the Opus Imperfectum, Jansenius Barrada, Arnoldi, Schegg, Schanz, Fillion, Keil, Weiss, etc.] or from the beginning of v. 13 [Glossa Ordinaria, Alb., Dionysius]. For the direct address is also found in the following parts of the discourse addressed to the multitudes; and the comparison with the salt occurs again in Lk. 14:34, where our Lord does not confine himself to his disciples. The comparison of the light is quite common in the language of the New Testament [cf. Phil. 2:15; Eph. 5:8; 1 Thess. 5:5; Prov. 4:18; 1 Pet. 2:9], so that it is not peculiar to the apostles. On the other hand, we do not wholly agree with Aug. and Bed., who regard the passage as equally addressed to the multitudes and the disciples; for though both these classes were present, the subject matter is such that it describes the apostolic trials and labors more clearly than those of common Christians.
Mat 5:14 You are the light of the world. A city seated on a mountain cannot be hid.
Mat 5:15 Neither do men light a candle and put it under a bushel, but upon a candlestick, that it may shine to all that are in the house.
Mat 5:16 So let your light shine before men, that they may see your good works, and glorify your Father who is in heaven.
(B) They are to Guide Men. You are the light of the world. The world is not only steeped in moral corruption, but also in intellectual darkness; hence our Lord’s salvific influence must remedy the latter as well as the former. The prophets foretold this [Is. 42:6; 49:6; cf. Isa 9:2], and the gospels point to the fulfilment in Jesus Christ [Jn. 1:9; Jn 8:12]. St. Paul, too, repeatedly mentions the apostolic office of enlightening the world [Acts 13:47; Col. 1:24; Phil. 2:15; Eph. 5:8].
A city set on a Mountain cannot be hid. The apostles are not merely to reflect the light of their master, but they are also to occupy such a position in the world that their doctrinal influence is not interfered with by intervening objects (the image of the candle under a bushel, verse 15). It is immaterial whether Jesus in pronouncing this sentence pointed out (the city of) Saphet or the fortification on Mount Tabor as the city that cannot be hid [cf. Guérin, Galilée, ii. 425]; in any case, his language resembles that of Isaias 4:1; Isa 2:2 [cf. Chromatus, Bede, Rabanus Maurus, Chrysostom, Jansenius, Lam., Cajetan]. It follows from these words that the church must be always visible, since she is compared not merely to a city on a mountain, but to a city so placed on the mountain that it cannot be hid.
It is not only the church that must be visible to the world, but each individual Christian, too, must contribute his share in shedding the light of Jesus Christ. (If the city set on a hill cannot be hid, neither can its inhabitants). They are warned that not even the worldly-wise apply things to a purpose opposed to that for which they are destined: they do not light their candle and put it under a bushel, or the ordinary household measure holding about a peck, but on the lamp-stand, fastened in the wall, so that the light may be diffused as far as possible. The lamp was not extinguished during the night, but when its light was not desired for a space of time, it was placed under an inverted hollow cover. In the same manner, our Lord wishes his light burning in the church, to illume the darkness of the world [Bruno, Jansenius, Maldonado].
But our light must not shine before the world indiscriminately: two conditions are needed to render the shining of one’s light advisable. [a] The light of doctrine must be accompanied by good works, because principles without practice will be of little avail. This is confirmed by the fact that St. Paul found it necessary to appeal repeatedly to his good works: 2 Cor. 11:16 ff.; 1 Pet. 2:12 [Bruno, Chromatus, Jansenius]. [b] Our light must so shine before men that they may glorify our Father who is in heaven. It is not our own glory that must be the object of showing our light before the world. Here again we have St. Paul as our model [cf. Gal. 1:24]. That the glory of God must be the end of all our actions, our Lord emphasizes especially by the name Father, reminding us that we must glorify God as good sons glorify their parents [cf. Cajetan, Augustine]. God is repeatedly called Father in the Old Testament [Is. 63:16; Deut. 32:6; Wisd. 2:16; 14:2; Sirach 23:1; Sirach 51:10; Tob. 13:4]; but the name Father as opposed to the other two persons of the Holy Trinity is peculiar to the New Law. Even the Rabbis hold that God is glorified by the good works of men, and dishonored by their evil deeds [Schöttgen], though they apply these principles only to our private life [cf. R. Bechonya, Bahir f. 5, 3; Soh. Lev. f. 2, 5].