Let's analyze the mathematical questions from the uploaded image.

1. Continuity and Differentiability of the given function:

• The first part asks you to examine the continuity and differentiability of the function $f(x) = |x - 1| + |x - 2|$ on the interval $[1, 2]$.
  • Continuity: A piecewise absolute function like this is continuous on $[1, 2]$, since the absolute value functions are continuous everywhere.
  • Differentiability: To check differentiability, we need to see if there are any points where the derivative doesn't exist (typically where $x = 1$ or $x = 2$, since the function changes behavior at these points).

2. Euler's Theorem Verification:

• Problem (a): Verify Euler's theorem for: $u = x^2 \tan^{-1}\left(\frac{y}{x}\right) - y^2 \cos^{-1}\left(\frac{x}{y}\right)$
  • Euler's theorem involves homogeneous functions and partial derivatives.
  • The objective here is to verify if Euler's theorem holds for this function.

3. Prove Independence of $u, v, w$:

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