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definition of a derivative 

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f'(x)=lim h0 (f(x+h)f(x))/h provided that the limit exists. 


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the mathmatical definition of continuity 

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a function y=f(x) is continuous at a point c of its domain if: lim xc f(x)=f(c) 


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if f(u) is differentiable at the point u=g(x) and g(x) is differentiable at x, then the composite function f(g(x)) is differentiable at x, and: f(g(x))=f'(g(x))*g'(x) 


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if f is continuous on a closed interval [a,b],then f attains both an absolute max value M and an absolute minimum value m in [a,b]. That is,there are numbers x1 and x2 in [a,b] with f(x1)=m, f(x2)=M, and f(x) is between m and M for every other x in [a,b]. 


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if f has a local max or minimum value at an interior point c of its domain, and if f' is defined at c, then f'(c)=0 


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suppose y=f(x) is continuous on a closed interval [a,b],and differentiable on the interval's interior (a,b) at which f(b)f(a)/ba = f'(c) 


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