Continuity and Differentiability: True or False Statements

Continuity and Differentiability: True/False

1. 1. If the function g(x) and f(x) are both continuous, then the function g(x) + f(x) is discontinuous. Answer: False.
2. 2. If f(x) is not differentiable at x = a, then f(x) is discontinuous at x = a. Answer: False.
3. 3. If the function g(x) is discontinuous, then there does not exist a function f(x) such that g(x)/f(x) is continuous. Answer: False.
4. 4. If a continuous curve is smooth at a point, then a straight line is seen in the infinite zoom-in scope. Answer: True.
5. 5. If f(x) is differentiable at x = a, then f(x) is rough at x = a. Answer: False.
6. 6. If, with f(x) ≠ 0, the function g(x) f(x) is discontinuous for any discontinuous function g(x), then the function f(x) ≠ 0 is discontinuous. Answer: False.
7. 7. If the limit lim_x→a f(x) does not exist, then f(x) is not differentiable at x = a. Answer: True.
8. 8. If the function g(x) f(x) is discontinuous, then g(x) and f(x) are not both continuous. Answer: True.
9. 9. If the function g(x) > 0 is discontinuous, then there does not exist a function f(x) such that f(x) g(x) is continuous. Answer: False.
10. 10. If f(x) is not differentiable at x = a, then the limit lim_x→a (x−a)⁻¹(f(x) − f(a)) does not exist. Answer: True.
11. 11. If g(x) > 0 and f(x) are both continuous, then the function g(x) f(x) is discontinuous. Answer: False.
12. 12. If, for the function g(x), there exists a function f(x) such that f(x) + g(x) is continuous, then the function g(x) is continuous. Answer: False.
13. 13. If the limit lim_x→a f(x) does not exist, then f(x) is discontinuous at x = a. Answer: True.
14. 14. If f(a) is defined, then f(x) is continuous at x = a. Answer: False.
15. 15. If, with f(x) ≠ 0, the function g(x) f(x) is continuous for some positive discontinuous function g(x) > 0, then the function f(x) ≠ 0 is discontinuous. Answer: True.
16. 16. If the function g(x) ≠ 0 is continuous, then there exists a function f(x) such that f(x)/g(x) is continuous. Answer: True.
17. 17. If the function g(x) is continuous, then there exists a function f(x) > 0 such that f(x) g(x) is continuous. Answer: True.
18. 18. If the function g(x) + f(x) is discontinuous for any discontinuous function g(x), then the function f(x) is discontinuous. Answer: False.
19. 19. If, for the function g(x), there exists a function f(x) such that f(x) g(x) is continuous, then the function g(x) is continuous. Answer: False.
20. 20. If the limit lim_x→a f(x) does not exist, then f(a) is undefined. Answer: False.

Additional statements:

If a function has a derivative at a certain point, then the function must be continuous at that point: f'(x) is continuous at x = a. Answer: False.

The function f(x) = |x| is not differentiable at x = 0, so it is also discontinuous at that point: f(x) = |x|. Answer: False.