Fundamental Theorem of Calculus for definite integrals... assume continuity?

So here's the problem:

I can evaluate the indefinite integral:

Integrate[D[u[x], x], x]

u[x]

However, I'd like to evaluate:

Integrate[D[u[x],x], {x, x0, x1}]

and get

u[x1] - u[x0]

Or especially, evaluate

Integrate[D[u[x, y], x], {x, x0, x1}]

and get

u[x1, y] - u[x0, y]

Is there a way that I can assume that D[u[x], x] is continuous in the range x0 to x1? Is there a some assumption that can be met in order for me to evaluate the fundamental theorem of calculus?

Comments

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What is and isn't a valid variable specification for Manipulate?

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