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One elementary application to ODEs: a differential equation $$u'(t)=f(t, u(t ))$$ with a continuous RHS of the form

$$f(t,x):=-\frac{\partial_t g(t,x)}{\partial_x g(t,x)},$$ for some $C^1$ function $g$ with non-vanishing $\partial_x g,$ has a one-parameter family of solutions whose graphs are the level sets of $g$. This is a largely used solution method, but it's the IFT that ensures that these solutions are indeed well-defined functions, implicitly defined by $g(t,u(t))=c.$ Incidentally, note that this is a case where there is unicity holds uniqueness for the Cauchy problem, even if the Lipschitz assumptions may fail to hold.
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One elementary application to ODEs: a differential equation $$u'(t)=f(t, u(t ))$$ with a continuous RHS of the form

$$f(t,x):=-\frac{\partial_t g(t,x)}{\partial_x g(t,x)},$$ for some $C^1$ function $g$ with non-vanishing $\partial_x g,$ has a one-parameter family of solutions whose graphs are the level sets of $g$. This is a largely used solution method, but it's the IFT that ensures that these solutions are indeed well-defined functions, implicitly defined by $g(t,u(t))=c.$ Incidentally, note that this is a case where there is unicity for the Cauchy problem, even if the Lipschitz assumptions may fail to hold.
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One elementary application to ODEs: a differential equation $$u'(t)=f(t, u(t ))$$ with a continuous RHS of the form

$$f(t,x):=-\frac{\partial_t g(t,x)}{\partial_x g(t,x)},$$ for some $C^1$ function $g$ with non-vanishing $\partial_x g,$ has a one-parameter family of solutions whose graphs are the level sets of $g$. This is a largely used solution method, but it's the IFT that ensures that these solutions are indeed well-defined functions, implicitly defined by $g(t,u(t))=c.$