Isaiah is one of several books in the Bible whose authorship has been attacked and denied by educated individuals. These individuals have put forth the idea that Isaiah did not write the entirety of the book of Isaiah, but instead propose one or two others added to his writings beginning with chapter 40, extending to the end of the book. Other books of the Bible have also been attacked in this manner, including the Pentateuch, Daniel, and the Gospels.
The claim I have heard from proponents of these philosophies is that to change the author does not defeat the inspiration of the Scriptures, nor does it harm the message of the Bible. In truth, the person who believes Isaiah did not write anything beyond Isaiah 39 must deny the Scriptures not only in Isaiah but also in several New Testament books as well. Consider these New Testament passages which specifically attribute the authorship of Isaiah 40, and beyond, to Isaiah the prophet:
Matthew 3:3 (Isaiah 40:3)
Matthew 12:18, 21 (Isaiah 42:1)
Luke 4:17 (Isaiah 61:1)
John 1:23 (Isaiah 40:3)
Acts 8:28 (Isaiah 53:7)
Romans 10:16 (Isaiah 53:1)
Romans 10:20 (Isaiah 65:1)
These passages in the New Testament proclaim Isaiah the prophet as the author of Isaiah chapters 40-66. To deny Isaiah as the inspired author of the book of Isaiah requires the skeptic to also deny the inspiration of each of the aforementioned New Testament Scriptures. These Scriptures become false, they proclaim lies, they are inaccurate if Isaiah is not the inspired author of Isaiah 40-66.
If the Scriptures proclaim falsehood as truth, they cannot be trusted, and furthermore they have not originated from the God who cannot lie (Titus 1:2). A person who denies Isaiah's authorship has no reason to believe any of the Scriptures. Instead, they have every reason to leave the Bible behind as just one more ancient book among several that exist presently.
The choice to accept or reject the Bible’s claims of authorship directly affects the entire collection of the Scriptures. If they originated with God, they cannot proclaim lies. If they did not originate with God, of what use are they at all?
 “Who Wrote the Book of Isaiah?,” Zondervan Academic, December 6, 2018, https://zondervanacademic.com/blog/who-wrote-isaiah.