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Continuity of Functions

Continuity of Functions

Continuity of Functions: Is It Discontinued? True or False

1.
  • Identify the picture(s) in which we can see what f is doing for all values of x in the window.

I. y = 5x 3– 6x2 with -0.4 ≤ x ≤ 0.4 

II. y = 5x3– 6x2 with -0.6 ≤ x ≤ 0.6  -> Both I and II

2. Four different graphs of the function f(x) = 2x2 + 1 are shown below. Which picture best illustrates the fact that if x is within 0.06 of 2, then f(x) is within 0.5 of f(2) = 9? ->


3. Which statement is true for the function f shown below? 

-> If x is within 0.5 of 3 then f(x) is within 0.5 of 1.

4. We have a function f. We want f(x) to be within 0.5 of f(0). For which of the following functions do we have a guarantee that we can restrict x to find what we want? (x must be allowed to move the same distance from 0 in either direction, and x may not just be set equal to 0.)

-> II and III

5. If f(x) = 4 – 2x then when |x – 1| < 0.5 we have a guarantee that |f(x) – 2| < 1. Identify c, ε, and δ as commonly used in the definition of continuity.

-> c = 1, ε = 1, δ = 0.5


6. Let f(x) = 3x + 1. Then f is continuous at 1 with f(1) = 4. Find the largest value of δ for which if x is within δ of 1, then f(x) is within 0.5 of 4.

-> 0.16

7. Let  Then f is continuous at 4 with f(4) = 2. Find the largest value of δ for which if x is within δ of 4, then f(x) is within 0.25 of f(4). -> 0.5
8. In the formal definition of continuity at x = c,

-> ε describes how close we want f(x) and f(c), while δ describes how close x and c need to be.

9. If the function f is continuous at c, then -> for every ε > 0 we can find a δ > 0 such that if |xc| < δ then |f(x) – f(c)| < ε.


10.
  • Let , c = 0, and ε = 0.1. 

Which is the largest value of δ such that if |xc| < δ, then |f(x) f(c)| < ε ? ->



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