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To follow up on the answer of @VARulle let me give you some conditions for which the indifference curve is path connected. The argument can also be found in the book Mathematical Methods and Models for Economists by Angel de la Fuente. Preferences are monotone if $x > y$ implies $x \succ y$ and that preferences are continuous if $x_n \succeq y_n$, $x_n \... 7 Given your last comment above it seems that what you are really asking is whether the indifference sets of a continuous preference relation on$\mathbb R^n_+$are path-connected. The answer is No. Let$n=1$and let the preference relation be represented by$u(x)=(1-x)^2$. Then the indifference set e.g. for$u=1$is$\{0\}\cup\{2\}\$, which is not path-...