To avoid undue hardship, nothing in this Part shall be deemed to require a change in the plans, construction or designated use of any building on which actual construction was lawfully begun prior to the effective date of adoption or amendment of this Part and upon which actual building construction had been carried on diligently. Actual construction includes the placing of construction materials in permanent position and fastened in a permanent manner. Where demolition or removal of an existing building has been substantially begun preparatory to rebuilding, the demolition or removal shall be deemed to be actual construction, provided that the work is carried out diligently.