You suggested Titus 2:11 For the grace of God has appeared, bringing salvation to all men,"
Interestingly, I have seen this verse used to defend "universal reconciliation", which completely un-scriptural.
Compare with Titus 3:4-7 4 But when the kindness and love of God our Savior appeared,
5 he saved us, not because of righteous things we had done, but because of his mercy. He saved us through the washing of rebirth and renewal by the Holy Spirit,
6 whom he poured out on us generously through Jesus Christ our Savior,
7 so that, having been justified by his grace, we might become heirs having the hope of eternal life.
Is faith in Christ a righteous thing that we do?
“Is faith in Christ a righteous thing that we do?”
“Therefore they said to Him, “What shall we do, so that we may work the works of God?”
Jesus answered and said to them, “This is the work of God, that you believe in Him whom He has sent.”
In scripture, faith is never referred to as a “work”. Neither is saving faith ever referred to as a “gift”. Faith is contrasted with work, but is is, by definition, something we do in response to God.
“primarily, “firm persuasion,” a conviction based upon hearing (akin to peitho, “to persuade”), is used in the NT always of “faith in God or Christ, or things spiritual.”
The word is used of (a) trust, e.g., Rom. 3:25 [see Note (4) below]; 1 Cor. 2:5; 15:14,17; 2 Cor. 1:24; Gal. 3:23 [see Note (5) below]; Phil. 1:25; 2:17; 1 Thess. 3:2; 2 Thess. 1:3; 3:2; (b) trust-worthiness, e.g., Matt. 23:23; Rom. 3:3, RV, “the faithfulness of God;” Gal. 5:22 (RV, “faithfulness”); Titus 2:10, “fidelity;” (c) by metonymy, what is believed, the contents of belief, the “faith,” Acts 6:7; 14:22; Gal. 1:23; 3:25 [contrast Gal. 3:23, under (a)]; Gal. 6:10; Phil. 1:27; 1 Thess. 3:10; Jude 1:3,20 (and perhaps 2 Thess. 3:2); (d) a ground for “faith,” an assurance, Acts 17:31 (not as in AV, marg., “offered faith”); (e) a pledge of fidelity, plighted “faith,” 1 Tim. 5:12.
The main elements in “faith” in its relation to the invisible God, as distinct from “faith” in man, are especially brought out in the use of this noun and the corresponding verb, pisteuo; they are (1) a firm conviction, producing a full acknowledgement of God’s revelation or truth, e.g., 2 Thess. 2:11,12; (2) a personal surrender to Him, John 1:12; (3) a conduct inspired by such surrender, 2 Cor. 5:7. Prominence is given to one or other of these elements according to the context. All this stands in contrast to belief in its purely natural exercise, which consists of an opinion held in good “faith” without necessary reference to its proof. The object of Abraham’s “faith” was not God’s promise (that was the occasion of its exercise); his “faith” rested on God Himself, Rom. 4:17,20,21.”
http://www2.mf.no/bibelprog/vines?word=%AFt0000987
Please note in this small sample, faith is something men do, often poorly, and not something God places in a person:
Mat 6:30
“But if God so clothes the grass of the field, which is alive today and tomorrow is thrown into the furnace, will He not much more clothe you? You of little faith!
Mat 8:10
Now when Jesus heard this, He marveled and said to those who were following, “Truly I say to you, I have not found such great faith with anyone in Israel.
Mat 8:26
He *said to them, “Why are you afraid, you men of little faith?” Then He got up and rebuked the winds and the sea, and it became perfectly calm.
Mat 9:2
And they brought to Him a paralytic lying on a bed. Seeing their faith, Jesus said to the paralytic, “Take courage, son; your sins are forgiven.”
Mat 9:22
But Jesus turning and seeing her said, “Daughter, take courage; your faith has made you well.” At once the woman was made well.
Mat 9:29
Then He touched their eyes, saying, “It shall be done to you according to your faith.”
Mat 14:31
Immediately Jesus stretched out His hand and took hold of him, and *said to him, “You of little faith, why did you doubt?”
Mat 15:28
Then Jesus said to her, “O woman, your faith is great; it shall be done for you as you wish.” And her daughter was healed at once.