As per the provisions of the Contract Act, a contract may be in writing, or it may be verbal, or it may in fact even be implied contract. No specific format is prescribed for a contract. A contract is basically a proposal and its acceptance.
As far as Contract Act is concerned, there is no specific requirement of a formal employment contract being signed between the employer and employee, or of any formal appointment letter being issued.
In the situation described in your question, if an employee joins a firm on the basis of an offer of employment and gives joining report, and signs attendance register everyday, it would be considered as an employment contract or agreement.
At the same time, while it is not binding or mandatory, it may be desirable that such a contract should be in writing or in a formal form to avoid any future disputes.
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