how the mode value is open in LhL and rhL in continity when h tends to zero

When h →0, it means h is not equal to zero but it is very near to zero.

 

For LHL we take h→0¯, it means that h is nearer to zero from left side i.e. h is less than zero.

For RHL we take h→0+, it means that h is nearer to zero from right side i.e. h is greater than zero.

The function |x| is defined as

     |x| = x  if x ≥ 0

or,  |x| = —x  if x < 0

In case of LHL when h→0¯, take |x| = —x.

and in case of RHL when h→0+, take |x| = x.