Question

f(x, y)=\(\frac{3xy+3y+3x}{5x+5y}\) continuous at origin.

(a) True

(b) False

The question was posed to me during an online interview.

My question is taken from Continuity topic in portion Complex Function Theory of Engineering Mathematics

Answer 1

Right option is (b) False

The best I can explain: Case 1: \(\lim_{x \to 0 \\ y \to 0}\frac{3xy+3y+3x}{5x+5y}\)

\(\lim_{y \to 0}\frac{3y}{5y}\)=\(\frac{3}{5}\)

Case 2: \(\lim_{y \to 0 \\ x \to 0}\frac{3xy+3y+3x}{5x+5y}\)

\(\lim_{x \to 0}\frac{3x}{5x}\)=\(\frac{3}{5}\)

Case 3: Along a linear path, \(\lim_{y \to mx \\ x \to 0}\frac{3xy+3y+3x}{5x+5y}\)

\(\lim_{x \to 0}\frac{3mx^2+3mx+3x}{5x}\)=\(\frac{3(m+1)}{5}\)

Since, the cases are not equal, f(x, y) is discontinuous at x=0.