AMU Engineering 2017 solved paper Solved Paper

The function f(x)=x−[x] is discontinuous at all integral points.
For this, let c be an integer.
Then, [c−h]=c−1
[c+h]=c and [c]=c
Now, LHL =

lim

x→c

f(x)=

lim

x→c

(x−[x])
=

lim

h→0

{(c−h)−[c−h]}
=

lim

h→0

(c−h)−(c−1)
=

lim

h→0

(c−h−c+1)=1
Also, RHL =

lim

x→c+

f(x)=

lim

x→c+

(x−[x])
=

lim

h→0

{(c+h)−[c+h]}
=

lim

h→0

(c+h−c)=0
∴LHL≠RHL
Thus, f(x) is discontinuous at all integers.
Hence, f(x) is continuous at non-integer points only.