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Consider the function f: R → R given by
f(x) = 5, for x ≤ 1;
a + bx, for 1 < x < 3;
b + 5x, for 3 ≤ x < 5;
30, for x ≥ 5.
Which of the following is true about f?
- f is continuous when a = 5 and b = 5
- f is continuous when a = −5 and b = 10
- f is continuous when a = 0 and b = 5
- f is never continuous, whatever the values of a and b
Correct answer: f is never continuous, whatever the values of a and b
Solution
Continuity needs a+b=5 (at x=1), a+2b=15 (at x=3) and b=5 (at x=5). The last two give a=5, but then a+b=10 != 5, a contradiction. No a,b make f continuous, so f is never continuous.
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