Let f : R → R be a function defined as . Then f is

Engineering

Mathematics

Basic Definition of Continuity

Piecewise Defined Functions

domain of function

Question

Let f : R → R be a function defined as $f (x) = {\begin{cases} 5, if x \leq 1 \\ a + bx, if 1 < x < 3 \\ b + 5 x, if 3 \leq x < 5 \\ 30, if x \geq 5 \end{cases}$ . Then f is

Continuous if a = – 5 and b = 10

Continuous if a = 5 and b = 5

Continuous if a = 0 and b = 5

Not continuous for any values of a and b

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For x = 1

R.H.L = a + b

L.H.L = 5

So to be continuous at x = 1

a + b = 5 …………(i)

for x = 3

R.H.L. = b + 15

L.H.L = a + 3b

b + 15 = a + 3b

a + 2b = 15 ………….(ii)

for x = 5

R.H.L = 30

L.H.L = b + 25

b + 25 = 30

b = 5.

From equation (ii)

a = 10

but a = 10 and b = 5 does not satisfied equation (i)

So f (x) is discontinuous for a $\in$ R and b $\in$ R