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Question:

Let f:R→R be a function defined as:
f(x) =
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩5, if x≤1
a+bx, if 1<x<3
b+5x, if 3≤x<5
30, if x≥5
Then, f is

Continuous if a=-13 and b=10

Not continuous for any values of a and b

Continuous if a=5 and b=5

Continuous if a=0 and b=5

Solution:

f(x) = ⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩5, if x≤1
a+bx, if 1<x<3
b+5x, if 3≤x<5
30, if x≥5
f(1) = 5, f(1-) = 5, f(1+) = a+b
f(3-) = a+3b, f(3) = b+15, f(3+) = b+15
f(5-) = b+25; f(5) = 30 f(5+) = 30
From above we concluded that f is not continuous for any values of a and b.