f(x) = 9 and f(x) = 9, so f(x) = 9, which satisfies condition 2.
f(x) = 9 = f(2), which satisfies condition 3. Therefore, f(x) is continuous at x = 2.
- No. It fails condition 3.
In order for a function f(x) to be continuous at a point x = c, it must fulfill all three of the
following conditions:
Condition 1: f(c) exists.
Condition 2: f(x) exists.
Condition 3: f(x) = f(c).
Let’s test each condition.
f(3) = 29, which satisfies condition 1.
f(x) = 30 and f(x) = 30, so f(x) = 30, which satisfies condition 2.
But f(x) ≠ f(3). Therefore, f(x) is not continuous at x = 3 because it fails condition 3.
- No. It is discontinuous at any odd integral multiple of .
Recall that sec x = . This means that sec x is undefined at any value where cos x = 0,
which are the odd multiples of . Therefore, sec x is not continuous everywhere.
- No. It is discontinuous at the endpoints of the interval.
Recall that sec x = . This means that sec x is undefined at any value where cos x = 0.
Also recall that cos = 0 and cos = 0. Therefore, sec x is not continuous everywhere
on the interval .
- Yes.
