Yves,

For your categorical dependent variable, how many levels (categories) are there?  If there are more than 2, you can consider using an ordinal or multinomial logistic regression model.

Mike

Hi Michael,

Thank you for your suggestion,

Actually it is two level only ,

bivariate non linear association study.

Ok, this is actually a little better.  You can use a standard logistic regression model with your continuous variable as the predictor.  Since your categorical outcome has two levels, you'll just need to specify which is the outcome of interest and which is the reference.  Some programs require you to re-code your categorical variables as a 0/1 binary variable with 1 as the outcome and 0 as the reference.  

Mike

Thank you so much  Michael,

I will look into it.

Yves,

You can also look at Maximal correlation. It is also sometimes used as a measure to capture non-linear associations.

Thank you Emmanuel,

I will look into it.

Thank you All for your suggestions,

I finally used Mutual Information using permutation test to do the association analysis.

I am very grateful for all your inputs.