Answer:
Let's analyze each function separately to find the values of x for which they are continuous:
(a) f(x) = {x^2 + 10 if x ≠ 0, 0 if x = 0}
To check for continuity at x = 0, we need to evaluate the limit as x approaches 0 from both the left and the right and see if they match the function value at x = 0.
Limit as x approaches 0 from the left:
lim (x → 0-) (x^2 + 10) = 10
Limit as x approaches 0 from the right:
lim (x → 0+) (x^2 + 10) = 10
Function value at x = 0:
f(0) = 0
Since the limit from both sides matches the function value at x = 0, the function is continuous at x = 0.
(b) f(x) = x^2 - (2x / (x^2 - 3x + 2))
To check for continuity, we need to find the values of x that make the denominator zero (potential discontinuities) and then check the limit as x approaches those values.
The denominator is x^2 - 3x + 2, which factors as (x - 1)(x - 2).
Setting the denominator equal to zero and solving for x:
x - 1 = 0 => x = 1
x - 2 = 0 => x = 2
Potential discontinuities occur at x = 1 and x = 2. Let's check the limits at these points:
Limit as x approaches 1:
lim (x → 1) [x^2 - (2x / (x^2 - 3x + 2))] = 1^2 - (2 * 1 / (1^2 - 3 * 1 + 2)) = 1 - (2 / (1 - 3 + 2)) = 1 - (2 / 0) (undefined)
Limit as x approaches 2:
lim (x → 2) [x^2 - (2x / (x^2 - 3x + 2))] = 2^2 - (2 * 2 / (2^2 - 3 * 2 + 2)) = 4 - (4 / (4 - 6 + 2)) = 4 - (4 / 0) (undefined)
The function is not continuous at x = 1 and x = 2 due to the division by zero in the denominator.
(c) g(x) = |x + 1|
The absolute value function is continuous everywhere, so g(x) is continuous for all real numbers.
(d) f(x) = {x^2 if x ≤ 1, x - 1 if x > 1}
To check for continuity, we need to evaluate the limit as x approaches 1 from both the left and the right and see if they match the function value at x = 1.
Limit as x approaches 1 from the left:
lim (x → 1-) (x^2) = 1^2 = 1
Limit as x approaches 1 from the right:
lim (x → 1+) (x - 1) = 1 - 1 = 0
Function value at x = 1:
f(1) = 1^2 = 1
Since the limit from the left is not equal to the limit from the right, the function is not continuous at x = 1.
In summary:
(a) f(x) is continuous for all real numbers.
(b) f(x) is not continuous at x = 1 and x = 2.
(c) g(x) is continuous for all real numbers.
(d) f(x) is not continuous at x = 1.
Explanation: