Paul says to "teach these things." What things? Everything he has just written, including even the difficult subject of definite atonement (see below). The "whole counsel of God" (Acts 20:27) is to be taught.
The negative commands here in 1 Timothy 4:12 and in 1 Timothy 2:14 may indicate that Timothy had a tendency towards shyness or timidity. Furthermore, some in the Church in Ephesus may not have accepted his authority because he was young. Timothy was probably in his thirties (under 40 was considered young in this culture - see Ireneus, Against Heresies, II. xxii) and therefore younger than many of the Christians and even some elders in Ephesus.
Paul told Timothy to "set an example ... in purity." Timothy was to establish his authority not by flaunting it, but by setting an example of godly living (Tit. 2:7). Milne says, "Timothy is to lead by example. This way he will stand the best chance of silencing his critics and the skeptics." See WLC 129. Kistemaker gives some definitions of the purity of which Paul speaks:
a. in speech, that is, in personal conversation (for preaching see the next verse).
b. in conduct, that is, in customs, habits, ways of dealing with people, etc.
c. in love, that is, in deep personal attachment to his brothers and in genuine concern for his neighbors (including even his enemies), always seeking to promote the welfare of all.
d. in faith, that is, in the exercise of that gift of God which is the root from which love springs (note: love here probably indicates the horizontal relationship; faith, the vertical).
e. in purity (see also 1 Tim. 5:2), that is, in complete conformity, both in thought and act, with God's moral law.
Let no one despise your youth. The negative command here may denote some timidity in Timothy (cf. 4:14; 2 Tim 1:7), but also the reality that some, even beyond the false teachers, may not have accepted his authority or mandate from Paul. Again, Paul urges both correction and a constructive example.
Instead, set an example for those who believe. Moral authority is gained and sustained by a good example in words, deeds and relationships. Indeed, this short summary echoes the character qualifications of church officers (see note 3:1-7 and 3:8-13).