Demonstrate, for a call price C and put price P, that if the forward given by put-call parity 

is the same for all K, then if this forward is used to back out an implied volatility from C, the same implied volatility must also be obtained from P. Explain why, on this basis, we would expect the implied volatility surface backed out from the riskless forward to be continuous, regardless of default.

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F= (CP)e + K