Original
Let {{MATH_8}}. The domain of {{MATH_9}} is
{{MATH_10}}.
{{CMD_11}}
{{CMD_12}}
{{CMD_13}}{{MATH_14}} is not everywhere continuous on {{MATH_15}}{{CMD_16}}
{{CMD_17}}{{MATH_18}} is continuous on {{MATH_19}} for some {{MATH_20}}{{CMD_21}}
{{CMD_22}}There is some {{MATH_23}} such that {{MATH_24}} is not continuous on {{MATH_25}}.{{CMD_26}}
{{CMD_27}}{{MATH_28}} is nowhere continuous on {{MATH_29}}{{CMD_30}}
{{CMD_31}}{{MATH_32}} is continuous on {{MATH_33}} for every {{MATH_34}}.{{CMD_35}}
{{CMD_36}}{{MATH_37}} is continuous on {{MATH_38}}{{CMD_39}}
{{CMD_40}}
{{CMD_41}}
The preceding statements regarding {{MATH_42}} are true because
{{CMD_43}}
{{CMD_44}}{{MATH_45}} is the product of two functions whose intervals of continuity have union equal to {{MATH_46}}.{{CMD_47}}
{{CMD_48}}the natural logarithm is not continuous.{{CMD_49}}
{{CMD_50}}the limit {{MATH_51}} exists and is equal to {{MATH_52}}.{{CMD_53}}
{{CMD_54}}{{MATH_55}} is the product of two functions which are continuous on {{MATH_56}}.{{CMD_57}}
{{CMD_58}}
{{CMD_59}}
Next, we shall classify the critical points of {{MATH_60}}.