What is the guarentee of existence of the differentiable function $t=\phi(t^{*})$ in the proof?

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What is the guarentee of existance of the differentiable function $t=\phi(t^{*})$ in the proof?

Reference:Advance Engineering Mathematics-Erwin Kreyzig[9 E][page:324] enter image description here]click here to enlarge image

How does two representation of the same curve relate by a differentiable function $\phi$? I don't understand this line, Rest of the line is fine for me. Please help me.