Is the Equation (1-x)Cosx = Sinx Continuous and Solvable in (0,1)?

[ken1101]

prove that the equation (1-x)Cosx=Sinx has at least one solution in (0,1)

I am having some problem in proving that the equation is continous.

Please help. Thank you

 

[mathman]

1-x , cosx, sinx are all continuous, and you are not doing any dividing, so continuity is obvious.

 

[ken1101]

I realize it is continous. I just need prove that it is continuous using the definition of continuous

for every ε > 0 there exists a δ > 0 such that for all x ∈ I,: |x-c|<δ⇒|f(x)-f(c)|<ε

 

[mathman]

Can you assume each of the three terms are continuous or do need to first prove that? Once that is done, it is straightforward for whole expression since absolute value of each term is ≤ 1.