Problem on contractible spaces

Question

Assignment:

Let X be a contractible space:

a) Show that X is path connected.
b) Show that any two continuous maps f , g Y→X  where Y is any topological space, are homotopic.
c) Let x₁ ∈ X and Cx₁ : X→X be the map defined by Cx₁ (x) = x₁  . Show that id_x  and Cx₁ are homotopic.

Provide complete and step by step solution for the question and show calculations and use formulas.