This Bible contradiction is from the Skeptic's Annotated Bible.
Yes, all of Saul's family died with him.
1 Chronicles 10:6 View context
So Saul died, and his three sons, and all his house died together.
No, one of Saul's sons (Ishbosheth) became king of Israel after Saul died.
2 Samuel 2:1-7 View context
And it came to pass after this, that David enquired of the LORD, saying, Shall I go up into any of the cities of Judah? And the LORD said unto him, Go up. And David said, Whither shall I go up? And he said, Unto Hebron.
So David went up thither, and his two wives also, Ahinoam the Jezreelitess, and Abigail Nabal's wife the Carmelite.
And his men that [were] with him did David bring up, every man with his household: and they dwelt in the cities of Hebron.
And the men of Judah came, and there they anointed David king over the house of Judah. And they told David, saying, [That] the men of Jabesh-gilead [were they] that buried Saul.
And David sent messengers unto the men of Jabesh-gilead, and said unto them, Blessed [be] ye of the LORD, that ye have shewed this kindness unto your lord, [even] unto Saul, and have buried him.
And now the LORD shew kindness and truth unto you: and I also will requite you this kindness, because ye have done this thing.
Therefore now let your hands be strengthened, and be ye valiant: for your master Saul is dead, and also the house of Judah have anointed me king over them.
No, David sacrificed two of Saul's sons and five of his grandsons to God to stop him from starving people to death.
2 Samuel 21:1-14 View context
Then there was a famine in the days of David three years, year after year; and David enquired of the LORD. And the LORD answered, [It is] for Saul, and for [his] bloody house, because he slew the Gibeonites.
And the king called the Gibeonites, and said unto them; (now the Gibeonites [were] not of the children of Israel, but of the remnant of the Amorites; and the children of Israel had sworn unto them: and Saul sought to slay them in his zeal to the children of Israel and Judah.)
Wherefore David said unto the Gibeonites, What shall I do for you? and wherewith shall I make the atonement, that ye may bless the inheritance of the LORD?
And the Gibeonites said unto him, We will have no silver nor gold of Saul, nor of his house; neither for us shalt thou kill any man in Israel. And he said, What ye shall say, [that] will I do for you.
And they answered the king, The man that consumed us, and that devised against us [that] we should be destroyed from remaining in any of the coasts of Israel,
Let seven men of his sons be delivered unto us, and we will hang them up unto the LORD in Gibeah of Saul, [whom] the LORD did choose. And the king said, I will give [them].
But the king spared Mephibosheth, the son of Jonathan the son of Saul, because of the LORD'S oath that [was] between them, between David and Jonathan the son of Saul.
But the king took the two sons of Rizpah the daughter of Aiah, whom she bare unto Saul, Armoni and Mephibosheth; and the five sons of Michal the daughter of Saul, whom she brought up for Adriel the son of Barzillai the Meholathite:
And he delivered them into the hands of the Gibeonites, and they hanged them in the hill before the LORD: and they fell [all] seven together, and were put to death in the days of harvest, in the first [days], in the beginning of barley harvest.
And Rizpah the daughter of Aiah took sackcloth, and spread it for her upon the rock, from the beginning of harvest until water dropped upon them out of heaven, and suffered neither the birds of the air to rest on them by day, nor the beasts of the field by night.
And it was told David what Rizpah the daughter of Aiah, the concubine of Saul, had done.
And David went and took the bones of Saul and the bones of Jonathan his son from the men of Jabesh-gilead, which had stolen them from the street of Beth-shan, where the Philistines had hanged them, when the Philistines had slain Saul in Gilboa:
And he brought up from thence the bones of Saul and the bones of Jonathan his son; and they gathered the bones of them that were hanged.
And the bones of Saul and Jonathan his son buried they in the country of Benjamin in Zelah, in the sepulchre of Kish his father: and they performed all that the king commanded. And after that God was intreated for the land.