The function f(x) = 4x + 1 is a line, therefore it's continuous everywhere. In particular, it's continuous at x = 5 with f(5) = 21. How must the x-values be restricted if we want to have 20.5< f(x) < 21.5?
Since we're looking at continuity near x = 5, 5 is our value of c and f(c) = f(5) = 21. Graph the function and set the window so that
0 ≤ x ≤ 100 ≤ y ≤ 42.
We do this so that (5,21) is in the center of the window. Now start narrowing the values of x, keeping 5 in the center, until the function goes out the sides of the graph instead of the top and bottom. Here's one possible progression, where we first bring x within 1 step of 5, then within 0.5 of 5, then within 0.25 of 5, and finally within 0.125 of 5:
If we restrict x so that 4.875 ≤ x ≤ 5.125,we find what we want: a picture where (5,21) is in the middle, and all the values of f(x) are between 20.5 and 21.5.