10. Sequences of Functions

Preservation Of Continuity — Quiz

Test your understanding of preservation of continuity with 5 practice questions.

Practice Questions

Question 1

Suppose each $f_n$ is continuous on $[a,b]$ and $f_n \to f$ uniformly on $[a,b]$. What can you conclude about $f$?

Question 2

Does pointwise convergence of continuous functions always preserve continuity of the limit?

Question 3

What is the pointwise limit of $f_n(x)=x^n$ on $[0,1]$?

Question 4

Why does $f_n(x)=x^n$ fail to converge uniformly on $[0,1]$ to its pointwise limit?

Question 5

If each $f_n$ is continuous on $[a,b]$ and $f_n \to f$ uniformly on $[a,b]$, what must be true at every point $c\in[a,b]$?