Mathematics - Online Test

The triangle formed by these lines is a right angled triangle

If the lines are perpendicular to each other, then the product of their slopes is -1

The slope of lines  3x + y – 4 = 0 , x - 3y – 4 = 0 are  -3 and 1/3 respectively.

The product of the slopes is -1

Hence these two lines are perpendicular to each other

This infers that the triangle formed by these lines is a right angled triangle.

Case 1 Let c be a real number which is not equal to any integer. for all real numbers close to c the value of the function is equal to [c]; i.e., limx→cf(x)=limx→c[x]=[c]. Also f(c)=[c]and hence the function is continuous at all real numbers not equal to integers.

Case 2 Let c be an integer. Then we can find a sufficiently small real number r>0 such that [c−r]=c−1whereas[c+r]=c

This, in terms of limits mean that limx→c−f(x)=c−1,limx→c+f(x)=c

Since these limits cannot be equal to each other for any c, the function is discontinuous at every integral point.